Is my understanding of the delayed choice quantum eraser correct? I'll say in advance that I am by no means an expert on Quantum Mechanics. I understand the basic mathematics of it (Wave function and Schrodinger equation), but did not go deeply into it or study it in a complete way.
I am intrigued by the lessons QM shows us about the nature of reality, and originally when reading about the double slit experiment I just assumed that the measurement by a device at one of the slits collapses the wave function.
Now I saw a video on YouTube (https://www.youtube.com/watch?v=SzAQ36b9dzs) which explains the delayed choice quantum eraser.
This video may be a simplification and not the actual experiment done. So I'll mention in short what is shown:


*

*It is the same configuration as the double slit experiment but with the addition of a "quantum eraser". We have two slits and we may or may not include detectors at the slits. We also have a "quantum eraser"; this is a device which can change detector results in such a manner that "which-way" information can no longer be attained from the detector.


I'd like to verify my understanding of this experiment and its conclusions (that it my question; whether I am understanding it correctly):


*

*if the information in the detectors is erased after detection by the detector, but before any observation by the experimenter (of either the detector or the screen), then we get an interference pattern, just as if the detectors were not there. This seem to suggest that whether we get an interference pattern depends on whether the "which-way" information "exists" (at the time of observation by the experimenter). 

*if the experimenter observes the detectors after detection but prior to deletion, we don't get an interference pattern (the wave function collapses); yet detection by the detectors, in itself, does not collapse the wave function (because as mentioned, we can still delete the information and get an interference pattern). This seems to suggest that observation (and perhaps other actions taken) by the experimenter is different in some way to detection by the detector.


So if I understand the experiment correctly, two main conclusions from it are those I indicated in bold text above. My question is whether my understanding and these conclusions are correct.
I originally wanted to include a question about my interpretation of these results in terms of the difference between the experimenter and the detector, but this is getting too long, so I'll leave it at that.
 A: The video that you watched on YouTube is full of gross errors and cannot be relied on to get an accurate picture of the quantum eraser experiment.
The underlying error in the video is that it claims there's detectors at the slits detecting which slit each photon traveled through. Nothing at the slits is detecting anything. What's placed before the slits in the experiment are nothing more than polarizing filters, typically one horizontal and one vertical. That's it! 
Almost two hundred years ago, Fresnel and Arago got together with a bunch of double-slit tools and accessories and discovered a remarkable fact about light: that exactly perpendicularly polarized light, when brought together via the double-slit, shows no interference. 
Well, according to the quantum explanation, what's really going on when such polarizers are placed over the slits in the double-slit experiment (i.e. which setup constitutes the quantum eraser experiment) is that each polarizer is, not detecting, but marking or imbuing each photon itself with distinctive information. (This is important because, in every instance except this, it's held by quantum mechanics that no two photons of the same wavelength can be distinguished.) But this information doesn't allow you to distinguish each and every photon; in particular, it allows you to mark each photon with only enough information to be able to tell, in theory, through which slit it traveled (i.e. because every photon traveling through each slit is imbued with the same information, that is, either its vertical or horizontal polarization.)
And you might be surprised to hear this, but never during the experiment does that "which-slit" information given to the photons have to be attempted to be read or deciphered. The point is solely that the photons were marked with this information. That, according to the quantum explanation, is what causes the absence of interference on the screen behind the slits. By marking the photons in this manner, the experimenter changes the behavior of the photons, causes them to relinquish their wave-like behavior, to choose only one slit through which to travel, and not to interfere on the other side.
It's also helpful to know that the "eraser" in the video is in reality nothing other than another polarizing filter rotated at 45 degrees from either of the two polarizing filters already present. Interposing this third filter, and as a result restoring the interference pattern on the screen behind the slits, is something that was also discovered by Fresnel and Arago. The point of this 'eraser' in the quantum eraser experiment is to show that one can change the behavior of the photons, i.e. restore their wave-like behavior by effectively erasing their which-way information, even after they've traveled through the slits exhibiting particle-like behavior.
Don't hesitate to let me know if you want any of the above clarified, if your questions based on the YouTube video still exist, or if you have any additional questions.
A: This is going to be a long answer because there is a lot to unpack here.
I'm going to briefly go over your questions about your understanding, then analyze the experiment from the video in detail, then explain why I think this experiment is not very interesting (since the video also tries to hype it).
Your understanding

if the information in the detectors is erased after detection by the detector, but before any observation by the experimenter (of either the detector or the screen), then we get an interference pattern, just as if the detectors were not there.

It doesn't matter when the "eraser" is used, or when the experimenter looks at anything; you get the same results regardless. Also, there's never an interference pattern visible on the screen. That pattern only shows up in later data analysis.

whether we get an interference pattern depends on whether the "which-way" information "exists" (at the time of observation by the experimenter).

That's correct in a certain sense. The way quantum mechanics works is that the system (as encoded in the wave function) interferes with itself as a whole. If there's which-path information anywhere in the system, then the paths are orthogonal, so they don't interfere.
But note that applying the "eraser" doesn't destroy the which-path information for this purpose, nor does any measurement, or anything else you can do without involving the photon. There is no way to destroy the which-path information unless you re-combine it with the photon, which doesn't happen in this experiment. "Quantum eraser" is really a misnomer.

detection by the detectors, in itself, does not collapse the wave function

That's correct of the "detectors" in this experiment. Typical thought-experiment "detectors" do collapse the wave function, though.

observation (and perhaps other actions taken) by the experimenter is different in some way to detection by the detector.

Observation by the experimenter is thermodynamically irreversible (and collapses the wave function), while the "detection" (by these detectors) is reversible (and doesn't). That distinction isn't important in this experiment, though. It's important that the "detection" at the slits be reversible, but the experimenter and the screen could just as well be reversible quantum computers. But it's probably easier, and just as correct, to think of them as wavefunction-collapsing classical objects.
The video (hype)
The video has some silly hype at the beginning:

[The experiment] demonstrates that the consequences of quantum mechanics are far stranger than we had ever imagined,

I disagree with this; I think that it's just a less interesting version of the EPR/Bell/Aspect experiment. I'll explain what I mean by this after the analysis.

and some people even claim that it indicates that messages are being sent backwards in time.

Those people are wrong. This isn't a matter of interpretation; they just don't understand how probability works. I'll come back to this after the analysis.
The video (post-hype)
I think (contra David Reishi) that the video is pretty accurate when it sticks to the physics, once you figure out what it means by "detector" and "quantum eraser".
In the video, the "detector" at each slit contains a two-state quantum system (a qubit) which is initially in some known state (say $|0\rangle$) and is flipped (to $|1\rangle$) if the photon passes through the detector. It might be quite hard to engineer such a device, but the laws of physics allow it: it's just a CNOT gate with the presence/absence of the photon as the control bit. (Arguably this shouldn't be called a detector or a measurement because it is reversible, but those are just words; it's clear what the device does physically.)
I'm going to simplify the setup a (qu)bit by omitting one of the detectors, because that doesn't lose any information: if the photon didn't go through that slit then it went through the other (in this idealized experiment free of engineering realities). The single detector is on the right slit, and gives us one qubit, which is $|0\rangle$ if the photon went through the left slit and $|1\rangle$ if it went through the right slit.
The "quantum eraser", which looks like a reject from a new-age crystal-healing video, is a quantum computer which simply applies a Hadamard gate to the qubit (that is, it takes $|0\rangle$ to $(|0\rangle+|1\rangle)/\sqrt2$ and $|1\rangle$ to $(|0\rangle-|1\rangle)/\sqrt2$).
The most serious error, or misleading statement, in the video is the implication that the "quantum eraser" erases which-path information, allowing the wave function to interfere again (13:04). That is not possible in quantum mechanics. There is nothing you can do to the qubit (or anything in the universe other than the photon), at any time, that will affect the observable behavior of the photon.
Analysis
Just after the "detection", the system is in the state $$|0\rangle|\text{photon in left slit}\rangle + |1\rangle|\text{photon in right slit}\rangle$$ (times $1/\sqrt2$; I'm going to ignore normalization factors for the most part). The photon then propagates to the screen. Just before the photon hits the screen, the state of the system is $$\sum_P (\alpha_P |0\rangle|\text{photon at P}\rangle + \beta_P |1\rangle|\text{photon at P}\rangle) = \sum_P (\alpha_P |0\rangle + \beta_P |1\rangle) |\text{photon at P}\rangle$$ where the sum is over all the points on the screen (all the pixels of the CCD, if you like).
If we had not put a detector at the slit, that sum would have been $$\sum_P (\alpha_P |\text{photon at P}\rangle + \beta_P |\text{photon at P}\rangle) = \sum_P (\alpha_P + \beta_P) |\text{photon at P}\rangle,$$ and the probability of finding the photon at P would therefore have been $|\alpha_P + \beta_P|^2$, which depends on the relative phase of $\alpha_P$ and $\beta_P$ (whose phases depend on the distances to the left and right slits respectively). But with the detector at the slit, the states scaled by $\alpha_P$ and $\beta_P$ are orthogonal, and so the probability of finding the photon at P is $|\alpha_P|^2 + |\beta_P|^2$, which does not depend on the relative phase. This is why the interference pattern disappears. In effect, there is interference only between identical states of the whole world, not just identical states of the photon we're detecting.
Suppose the photon is actually detected at P (which is now a specific point, not a bound variable of the summation). The state of the system (after wavefunction collapse) is now $\alpha_P |0\rangle + \beta_P |1\rangle$ times an overall normalization factor that we don't care about.
At this point we have a choice (the delayed choice): measure the qubit in the {0,1} basis, or apply the "quantum eraser" (Hadamard gate) and then measure the qubit in the {0,1} basis. In the former case, we'll get 0 and 1 with relative probabilities $|\alpha_P|^2$ and $|\beta_P|^2$ respectively. In the latter case, we'll get 0 and 1 with relative probabilities $|\alpha_P+\beta_P|^2$ and $|\alpha_P-\beta_P|^2$ respectively. (They're relative because of the normalization factors that I ignored.)
Now think about what happens if we condition (in the ordinary classical sense) on getting 0 or 1 in this measurement. If we didn't apply the Hadamard gate, $|\alpha_P|$ and $|\beta_P|$ just fall off monotonically with increasing distance from the respective slits, and they will be roughly equal, so we'll get 0 for around half the dots and 1 for the other half, across the board. If we did apply the Hadamard gate, $|\alpha_P \pm \beta_P|$ oscillate at the frequency of the interference pattern we would have gotten in the standard double-slit experiment. If P is near one of the peaks of $|\alpha_P+\beta_P|$, we'll almost certainly get 0; if it's near one of the troughs, we'll almost certainly get 1. So conditioning on Hadamard 0 or 1, we'll get an interference pattern.
Why the people who think this shows backward causation are wrong
You have a bag containing 4 balls, 2 red and 2 black. You draw a ball. There's a 1/2 chance it will be red. If it is red, there's a 1/3 chance that the second ball you draw will be red.
But if you don't look at the first ball, there's a 1/2 chance that the second ball you draw will be red, and if it is, there's a 1/3 chance that the first ball you drew was red. If you collect data over many trials, conditioned on the second ball being red, you'll find that indeed about 1/3 of the first balls you drew were red.
Backward causation? Of course not. If X is correlated with Y, then Y is correlated with X. It doesn't matter whether Y happened after X.
The argument for backward causation in the delayed-choice quantum eraser experiment is exactly the same as the argument for backward causation in this classical experiment.
Why this is a less interesting version of Bell's experiment
The initial state ($|0\rangle|\text{photon in left slit}\rangle + |1\rangle|\text{photon in right slit}\rangle$) is just a Bell state ($|0_A\rangle|0_B\rangle + |1_A\rangle|1_B\rangle$). There are three major differences between this experiment and the Bell experiment:

*

*The measurement axes range over the whole Bloch sphere (linear/elliptical/circular polarization for photons, or any spin axis for electrons), while in the Bell experiment they're restricted to a plane (linear polarization for photons, or a spin axis in some plane for electrons).


*The universe chooses the orientation of one of the detectors. (The dot on the screen encodes both the orientation and the result of the measurement, in this analogy.) For the other measurement we choose between two orientations (eraser or no eraser).


*The measurements of the two halves of the Bell pair are timelike separated, not spacelike separated.
The first difference doesn't matter much; I think you can still derive a version of the Bell inequality with this change. (If you can't, that's just another reason why this experiment is less interesting.)
The second and third differences make the experiment much less interesting, because they each independently make the whole thing consistent with a local hidden variable theory.
You can think of Bell's experiment as a game show similar to The Newlywed Game: the contestants are allowed to talk and agree on a strategy, then they're separated and independently asked questions that they didn't know in advance. Their goal is to give answers that are correlated in a certain sense.
If they're allowed to choose the question asked of contestant A, they can win easily, since they can agree on contestant A's question and answer in advance, and contestant B then has all relevant information when deciding how to answer. Likewise, if B's question is in the future light cone of A's question and answer, they can win if A surreptitiously sends that information to B by radio or dark matter or other light-speed-limited means.
You could replace my ball experiment with a classical simulation of the delayed-choice quantum eraser. For example: a computer generates two random numbers $x,y \in [0,1]$ and prints out two copies which are given to two oracles. The first oracle simply tells you the value of $x$. The second oracle, on your command, either tells you whether $y < \tfrac12$ or whether $y < \sin^2 100x$. If you condition on the answer to the first question, you get a flat histogram of $x$ values, while if you condition on the answer to the second question, you get a sinusoidal histogram. With minor tweaks to the second oracle's formulas, this simulation becomes exact.
Bell wrote an essay called "Bertlmann's Socks and the Nature of Reality" in which he presented a thought experiment similar to my four-ball experiment (involving his colleague Dr. Bertlmann who always wears mismatched socks), as an example of what the EPR experiment is not about. I think the people who invented the delayed-choice quantum eraser didn't understand the difference between quantum mechanics and Bertlmann's socks.
A: The trouble with interpretations of experiments such as the quantum eraser is that they are designed to challenge concepts or way of thinking that are deemed evident. They expose QM weirdness. Such experiments are not supposed to be easily interpreted; they're supposed to make us think deeper about our basic assumptions.
So, is your interpretation correct ? 
First of all let's say you will find a lot of people to think that it doesn't matter. What is correct is QM, and that is what the experiment shows. Maybe there is no need to interpret quantum mechanics; this is a nearly century-old debate.
Now your own interpretation here is based on the notion of information. But does information exists if it cannot be recorded ever? Is there a "which-way" information in the first place?
What I believe is not clear at all in your interpretation is that part:

if the experimenter observes the detectors after detection but prior
  to deletion, we don't get an interference pattern (the wave function
  collapses); yet detection by the detectors, in itself, does not
  collapse the wave function (because as mentioned, we can still delete
  the information and get an interference pattern). This seems to
  suggest that observation (and perhaps other actions taken) by the
  experimenter is different in some way to detection by the detector.

An "experimenter that observes a detector after detection" has to me the same semantics as just a "detection". You have had to phrase it this way in order to differentiate from a "detection by the detectors", which is where you assume some information is created that can further be erased. But to me if that information is to be erased, it just may have never existed at all, so "detection by the detectors, in itself" in that context means "no detection", actually. 
And so, no, observation by the experimenter is no different from detection by the detector; I don't think your conclusion is true. 
What is most interesting to me here is the logic of your thinking, and how you have been tricked by the name of the experiment: that it is called quantum eraser does not mean that this is what happens.
The trick is that if one believes there is always a specific path the photon travels, whether we detect it or not, then one cannot make sense of the double-slit experiment. Similarly, if you believe there is always a "which-way" information, whether we record it or not, then you cannot make sense of the quantum eraser experiment.
A: To understand this experiment, it is helpful to stop thinking in terms of the wave-particle duality and which-way. If you consider a photon to be a wave packet, or a packet of a superposition of waves, which describes reality much better, the experiment gets quite straightforward. See my video: https://m.youtube.com/watch?v=e-uopGrghFQ.
The bottom line is that it is not the presence of the which-way in itself that determines the appearance of the interference pattern, but the presence of a  phase shifting component (the beam splitter) or direct detection. The interference pattern itself is a post-selection effect: It is caused by the combination of the phase shift of one of the waves, resulting in a hit probability on the detectors that depends on the phase difference, thus the difference in path length, thus the angle at which the photon left the bbo-crystal, thus the location where it will end up on the detector screen.
