Is relativity of simultaneity an "observer issue"? There are some threads about this, but some answers seem to disagree.
First, this is what Einstein said on this matter:

The light rays emitted by the flashes of lightning A and B would reach him simultaneously", and again: " the observer will see the beam of light emitted from B earlier

and this is the reason why he says:

Events which are simultaneous with reference to him are not simultaneous with respect to the train, and vice versa.

(Source)
He basically says that it is the observation of two event that can either be simultaneous or not depending on the frame of reference.
But do the observers disagree on the simultaneity of the events after the "post-processing" (tracing back the actual events in spacetime)? It is at this point that things start to become difficult: after the "post processing", will the observers disagree on the order of the actual events, or will they agree?. Which is correct and why? And what does a Cauchy Hypersurface have to do with this?
 A: 
do the observers disagree on the simultaneity of the events after the "post-processing" (tracing back the actual events in spacetime)?

Yes. The relativity of simultaneity specifically refers to the disagreement about simultaneity that remains after the observers have accounted for the finite speed of light.
In other words, if I receive light from two novas tonight, one 100 light years distant and one 200 light years distant, I will say that they did not occur simultaneously. Similarly, if I receive the light from one 200 light years distant and read that a 100 light year distant nova was observed here 100 years ago, then I will say those did occur simultaneously.
Someone traveling with respect to me will make the same computations and disagree about both results.
A: It is better to think of the relativity of simultaneity as being a property of the geometry of spacetime, independent of any observers. Take this analogy... in 3D space there is no absolute 'up' direction. On Earth, we take 'up' by convention to mean a direction normal to local sea level, so 'up' in Australia is a quite different direction to 'up' in the US. We say two places are level if they have the same altitude, but that is only true in a given frame of reference. Sydney is above sea level in Australia, and New York is above sea level in the US, but New York is below sea level if you consider sea level to be an infinitely extended horizontal plane centred on Sydney, and Sydney is below sea level if you consider sea level to be an infinitely extended horizontal plane through New York. So the idea of 'level' is entirely frame-dependent.
In 4d spacetime, there is no absolute time direction. If you and I are moving relative to each other, our respective time axes are tilted. In your frame of reference, a plane of simultaneity is a level slice through spacetime normal to your time axis, which means that it is a sloping slice through time in my frame. That is like Sydney sea level being a sloping slice through space for a person in New York. It is just a property of reference frames. You don't need to bring observers into the picture. Indeed, mentioning observers in explanations of special relativity is, in my opinion, the cause of half of the misunderstandings that are raised by visitors to this site.
A: I want to address an ambiguity around the use of the word 'disagree' here.
I propose the following about thought demonstrations:
all of the protagonists featured in the narrative are to be physicists who are fluent in application of relativistic physics.
In a thought demonstration of relativity of simultaneity the two observers are both aware of relativity of simultaneity. Both observers will agree that the point of view of the other observer is self-consistent.
So: in that sense the two observers do not disagree with each other.


The actual message is: in Minkowski spacetime simultaneity is underdetermined.
As pointed out in a comment by stackexchange contributor Dale, the limits are that the same causality relations must still obtain.
Within the boundaries of still obtaining the same causality relations there is a certain leeway in adopting a plane of simultaneity.
Within that leeway Einstein synchronization procedure is one of the available options. For a given inertial coordinate system the Einstein synchronization procedure is the symmetrical approach. As a general rule, when there is leeway, pick the symmetrical choice as your convention.
(You do need to pick one choice, in order to represent physics taking place. Once you have made your pick - for a given inertial coordinate system - you must stick with that choice, otherwise you would introduce self-contradiction.)


To the two observers:
The two observers have a velocity relative to each other, so for each the co-moving inertial coordinate system is a different one. For each inertial coordinate system the Einstein synchronization procedure arrives at a different plane of simultaneity.
As stated earlier, we should assume the two protagonists of the thought demonstration are professional physicists, so they are aware of the reason why for two different inertial coordinate systems the Einstein synchronization procedure arrives at two different planes of simultaneity.
So that is why I find it awkward to see assertions that the two observers will disagree. Disagree about what? We should assume both are professional physicists: they will not disagree about the proper application of special relativity.


As pointed out in other answers: 'relativity of simultaneity' refers exclusively to that what remains after transmission delays have been taken into account.
(Therefore, if you read a discussion of relativity of simultaneity, and the author suggests that transmission delay effects are part of the story of relativity of simultaneity, then stop reading that author.)

Further viewing:
The series of animated gif's by Andrew Hamilton for special relativity, including discussion of relativity of simultaneity
A: The answer is yes only for the rather silly definition of "observer" (and "disagree") that is used in discussions of special relativity.
The "observers" in special relativity are not people or cameras or anything of the sort. They are networks of clocks and metersticks, large enough in space and time to reach whatever is being "observed" (spanning galaxies if necessary), and synchronized in a certain way. To "observe" an event means to note the position and time reading of the clock in that network that is nearest the event.
So when they say that two relatively moving observers have different notions of simultaneity, they are technically correct: two constellations of Einstein-synchronized clocks that are in relative motion will report different time readings where they coincide.
In reality, we don't construct coordinates in which we are personally at rest at all times. We use coordinate systems that are available to us. For example, on cosmological scales, the averaged Hubble flow can be used as a reference, giving what are called comoving coordinates. The solar system has a largish speed (about $0.001c$) in those coordinates. We don't define an alternate cosmological coordinate system in which we are at rest, because they would be inconvenient and would serve no purpose. On Earth, when you're driving to work, you normally think in coordinates affixed to the Earth, in which your workplace is stationary and you are moving toward it. It makes sense to define a car-centered coordinate system for some purposes, but those coordinates extend only as far as your car; it wouldn't make sense to extend them even to your workplace, much less to the Andromeda galaxy, because the car doesn't extend that far.
If driving speeds were large enough that time dilation noticeably affected wristwatch readings, people would adjust their wristwatches to match a reference clock when they arrived at work, because the whole point of having a wristwatch is to coordinate times with other people. They wouldn't claim that their personal wristwatch reading is the correct one and "disagree" with people who live a different distance from work.
I strongly dislike Dale's answer because it doesn't clarify the above points, but I like this comment he made:

In my opinion, the lesson to learn is that nature doesn’t care about simultaneity. It only cares about causality. Causes always come before effects, in all reference frames. But otherwise nature just doesn’t care.

"Observer" is a misnamed abstraction. The Earth and the Hubble flow are "observers"; you aren't. You can use any notion of simultaneity you want, as long as it respects light-cone causality. Nature doesn't care.
The relevance of Cauchy hypersurfaces is that in any global coordinate system that respects light-cone causality, the surfaces of constant time are Cauchy surfaces.
See also this answer.
