Relativity of simultaneity in special relativity I’ve read a lot of article about the topic and I think I understood it : according to the theory, two observers in two different frames of reference can disagree on the order of two events. For observer o1, event e0 happened before event e1, for observer o2, it is the reverse.
What bothers me though is that it seems to me it takes for granted that both observers look to the two events without knowing the basics of special relativity: if they knew better and had access to the other observer’s frame of reference, they would be able to see the reality of the events.
In the example of the Einstein train, if both the bystander and the train passengers know of the train movement, there is no reality relativity, there is an absolute state of the universe, right?
 A: 
In the example of the Einstein train, if both the bystander and the train passengers know of the train movement, there is no reality relativity, there is an absolute state of the universe, right?

Not in the meaning of the word "state" that you are thinking here. The issue is that there is no way for the bystanders and the train passengers to "know of the train movement". The relative velocity between the bystander and the passengers is a physical fact, but that relative velocity could be because the bystander is moving or because the passengers are moving or because both are moving. There is no possible way (even theoretically) to distinguish those cases. Therefore there is no way to set any absolute state of simultaneity because simultaneity depends on the reference frame.
There is an "absolute state of the universe", but simultaneity is simply not part of it. We typically don't use the word "absolute" to describe it, but instead use the word "invariant" or "covariant". "Absolute" has some bad connotations.
In the invariant description of the universe things are described in terms of coordinate-independent geometric objects called tensors. The tensors may be described with respect to some chosen basis, but they are themselves a geometric object that is independent of such descriptions.
Simultaneity is simply not a part of this tensor-based description of the universe. There is no "simultaneity tensor". The simultaneity concept itself is not part of the state of the universe in any invariant sense. Instead, what is invariant is causality. The universe "cares" that if A causes B then A must come before B, that is an invariant fact. But if A and B could not be causally related then the universe simply doesn't care which happens first. That idea that such non-causally-related events should have a temporal order is a human conceit, not a fact of nature.
A: In special relativity, the 'absolute state of the universe' is the Minkowski spacetime, where events are represented as points. In this spacetime, a point is four-dimensional, and there is no specific time dimension. Only within a frame of reference, which provides coordinates, are time and space separated.
So the notion of simultaneity does not make sense in the 'absolute state of the universe', because to compare times we first need a reference frame.
Relativity of simulaneity means that when simultaneity does make sense, that is with respect to a frame of reference, then the timewise ordering of events, in general, depends on that frame.
This is very counterintuitive, because we think that it is not possible to change the ordering of events - we immediately imagine that one event is somewhat the consequence or the cause of the other. But this is not the case: sufficiently separated events cannot effect one another, because the maximal speed of propagation of any consequence (the speed of light) is limited. So most event are in fact causally disconnected.
When we take this into account, using light cones to understand the causal structure of the Minkowski spacetime, then there is no problem anymore with the idea that two observers may see some events in different time orders.
A: ,You and I stand facing each other, looking for a coffee shop.  I say, "Oh, there's the coffee shop --- it's about a half a block to the left''.  Simultaneously, you say "Oh, there's the coffee shop --- it's about a half a block to the right''.
Do you want to say, on the basis of this interaction, that the two observers are unaware of some basic facts about the universe, and that if they had access to those facts, they would have a different view of reality?
Two observers facing each other apply different labels to the same direction in space.  Of course they can be completely aware of these differences, and the reasons for them, and still prefer to use their own labels.
Two observers in motion with respect to each other apply different labels to the same direction in spacetime.  One describes an event as taking place in the past; the other can describe the same event as taking place in the future.  Of course they can be completely aware of these differences, and the reasons for them, and still prefer to use their own labels.
Is there an absolute state of the universe?  Of course there is --- the coffee shop is where it is, and no place else.  Are there different ways to describe that state?  Of course there are.  You say the coffee shop is to the left and I say it's to the right.  We can perfectly well understand each others' perspectives and still prefer our own.  The existence of an absolute state does not mean that one of us has to be correct and the other incorrect.
Likewise, an explosion in Andromeda occurs as it occurs.  I describe that event as occurring in the past; you describe it as occurring in the future. We can understand each others' perspectives, still prefer our own, and recognize that neither perspective is better than the other.
This is not just some vague analogy --- it is an exact description of what's going on. Facing different directions in spacetime (which is what happens when we are in motion with respect to each other) leads us to choose different labels for the timing of events for exactly the same reason that facing different directions in space leads us to choose different labels for spatial directions.  Fully understanding that point is equivalent to fully understanding relativity.
A: It is very strange that relativity seems more obscure since its inception. Almost all ot the literature and the propounders or originators of the theory said that events are not simultaneous, while it is not correct though time is different for different frames in relative motion. To understand it needs a little patience. Relativity of simultaneity is case for effect on time when projectile is in direction of relative motion, here light is projectile. Before that, the development of this theory was for astrophysics and understanding of spectrometery of celestial bodies, the red and blue shift.
It was clear that there is no effect on time when projectile (in future refer it as light) or relative motion is perpendicular. As in swimmer analogy in stream, whether flow of stream in any direction, the time taken by swimmer is unaffected by direction and speed of stream in transverse direction of stream. So there's no effect on time or frequency of light from a body moving transversly to an observer, but its path or length is not remain unaffected but that not matter here. This also shows that speed of light is not invariant, but motion in perpendicular direction is.
This situation changes when light travels in the direction of the relative motion. The time taken by light is less if along and more if against the relative motion. One must know that light is not invariant to the relative motion and that causes doppler shift in phase of light. This shift help us to guess the distance once we know the relative speed. The case of relativity of simultaneity is similar to the case of longitudnally motion of light to the relative motion. Now focus is that as speed of light can be taken as invariant in transverse direction also in longitudnal motion. So it was devised that in spacetime, curve of motion is constant otherwise faster moving bodies (light in relative motion) curve less and slower moving bodies (light in rest frame) curve more. So we compensate for curvature by adding time to slower body or rest frame.
There is another issue that as speed of light is different for along and against the relative motion, so there is no single measure. But it was thought out that as in transverse direction, effect of relative motion is not observed on time, so if pythogorean theorem was used which compensate for speed by adding time in form of length, and as squared it remain invariant to along and against motion of light to the relative motion. In relativity of simultaneity, event is that moving observer sees two events simultaneous in its frame, so an observer on ground must see the events simultaneous, otherwise how laws of physics hold because both are reporting of two different events.
So concluding here, that observer on ground sees that light from backend of moving frame is traveling fast and front end is moving slow in comparison to ground, but an observer sees event simultaneous. So the event should be $S'_1$ and $S'_2$ for an observer with events frame and $S_1$ and $S_2$ for relative observer. Thus from theory of relativity's principle that no frame is preferred and speed of light is constant,$${S'}_1^2=c^2{t'}_1^2=S_1^2=c^2t_1^2+v^2t_1^2\\ \implies {t'}_1=\gamma t_1$$yes time booster is given to an object which is in proper frame. Similarly for ${t'}_2=\gamma t_2$. Here $v$ is relative speed and $\gamma=\sqrt{\frac{1}{1-\frac{v^2}{c^2}}}$.
But problem is that it conserved speed of light but booster is given once for faster object and then for slower object. On same ground length is contracted as $l=ct$. If booster is given on the basis of relative speed of object instead of relative speed of frame, we see no discrepancies in the observation of events and that which object is approaching or receeding is based on shift in frequency of light.If there is relativity of non-simultaneity and two events are not simultaneous in relative frames, then how comes Michelson and Morley didn't observer fringe shift. Either speed of light is not invariant or earth is not moving.
Use realtion of time in frequency shift as,$$f=\gamma f'\implies \delta f=(\gamma-1)f'=\frac{v^2}{2c^2}f'$$
