Does the Many Worlds Interpretation make any distinction between entangling processes and measurement? Suppose I have a qubit in the state $\alpha|0\rangle + \beta|1\rangle$. In the many worlds/relative state formulation of QM, if I measure this qubit with a measuring device in some initial state $|\text{initial}\rangle$, then, after measurement, the composite state of the qubit and measuring device is:
$$ \alpha|0\rangle|\text{measured 0}\rangle + \beta|1\rangle|\text{measured 1}\rangle.$$
In other words, the result of the measurement is to entangle the measuring device with the qubit.
In a second, unrelated experiment, suppose I have two qubits $q_0, q_1$ in the state $(\alpha|0\rangle + \beta|1\rangle)|0\rangle$. The application of a $\operatorname{CNOT}(q_0, q_1)$ gate to these two qubits transforms the state to:
$$ \alpha|0\rangle|0\rangle + \beta|1\rangle|1\rangle.$$
In both cases, the result to the wavefunction is the same - the second system is entangled with the first, and the only distinction here is the label I used to describe the kets in the second system. As a result, my questions are:

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*Do many worlds/relative state interpretations regard the two processes described above as different physically, or are they both regarded as "measurements"?

*If the two above processes are regarded as different, then, given a transformation of the form:
$$ (\alpha |\psi_1\rangle + \beta |\psi_2\rangle)|\phi\rangle \mapsto \alpha |\psi_1\rangle|\phi_1\rangle + \beta|\psi_2\rangle|\phi_2\rangle,$$
how can I tell whether a measurement has occurred or if this is just an entangling process (like the application of gates on a quantum computer)?

 A: Shortest answer: No!
Short answer: No, in the many worlds interpretation measurement is described as an interaction between a "system to be measured" and a "measurement system". In quantum mechanics it is generally true that an interaction between 2 systems can lead to entanglement.
Longer answer: No, the many worlds interpretation does not distinguish between the entanglement of two systems under some interaction and "measurement". Measurement is simply the interaction between a system that is being measured and the measurement system. In general, but not always, this interaction can lead to entanglement between two system.
Whether the interaction leads to entanglement depends on the initial states of both the system and the measurement device AS WELL AS the particular form of the interaction Hamiltonian. I make this distinction because, when thinking about the many worlds interpretation, it is important to make sure you are making basis independent statements. For example, the initial state of your qubit might be $(1/\sqrt{2})(|0\rangle + |1\rangle)$ which is a superposition state in the z-basis, but is not a superposition state in the x basis. So, while tempting, we can NOT state the following conclusion: "Measurement of a superposition state leads to entanglement between the measurement system and measurement device". Something closer to the case might be: "When a system is in a superposition of the measurement basis (which is determined by the form of the interaction between the measured system and measurement device) the measurement leads to entanglement between the two systems.
edit to address:

or are they both regarded as "measurements"?

Above I have claimed that in the Many Worlds Interpretation that measurement is no different from entanglement between the measurement device and measurement system.
But you seem to ask about the reverse. Is all entanglement a form of measurement?
The answer here is complicated, but, as I'll explain, doesn't really matter. What follows is my opinion. In my opinion, in the many worlds interpretation, a measurement arises when a measurement system interacts with a system to be measurement. The next question is then "what is a measurement system?" Intuitively there seems to be something at least qualitatively different between a qubit and a macroscopic photodetector. If we call the latter a measurement device and the former not a measurement device then, on my definition, entanglement with a photodetector DOES constitute a measurement while entanglement with a single qubit DOES NOT constitute a measurement.
This seems problematic because we are left with the problem that the Copenhagen interpretation faces which is "how do we define a measurement so that we know when the wavefunction collapses?" HOWEVER, there is a major difference. In the case of the Copenhagen interpretation the answer to the above question actually changes the rules of physics, whereas in the Many Worlds interpretation the answer to the question is purely semantic. The rules of physics do not change at all.
Above, I gave the opinion that entanglement with a qubit is not measurement while entanglement with a photodetector is. However, I'll state that, actually, in many cases, it CAN BE intuitively useful to think about entanglement with small quantum systems as a type of measurement. This is to say, in the Many Worlds interpretation, since it doesn't matter to the physics what is and what is not a measurement, we are free to move the definition of measurement (or the Heisenberg cut) around as much as we like. We can consider interaction with a qubit to BE a measurement or we can consider interaction with a human to NOT be a measurement. The physics is the same either way. This is the beauty and motivation of the many worlds interpretation.
Here are some of my (and your) other posts on the many worlds/everettian interpretation:

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*Is the many-worlds interpretation really just an interpretation?

*Does quantum mechanics tell us when the next measurement will occur?
Finally a disclaimer: This post sounds like a glowing approval of Many Worlds interpretation. While it is my favorite interpretation, it is not without shortcomings. The main shortcoming is that it doesn't not prescribe any way to determine subject mental states from physical quantum states. This makes it impossible or very hard to answer the question: "What does a human experience when his brain is in a superposition state of having experience incompatible outcomes?" But at this point I'm off topic.
