This seems to be a recurrent topic but I wasn't able to find any satisfactory discussion about this tought experiment.
I will quote the original story from Einstein's and put my doubts inside:
Up to now our considerations have been referred to a particular body of reference, which we have styled a “railway embankment.” We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated in Fig. 1. [...]
Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative.
When we say that the lightning strokes A and B are simultaneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A —> B of the embankment.
This definition of simultaneity seems indisputable for the embankment reference but what about the train reference? It seems to me that it would be suitable also for the train only if we assume invariance of the speed of light otherwise we should take into account the relative speed between the sources of light and the train. (It should be noted that the argument will not use invariance of speed of light later on so this seems to be the only possible place for this assumption).
So which is the physical theoretic model that we are using here? Classical mechanics? It wouldn't be consistent with the definition of simultaneity (as I said above). Maybe we are considering a hybrid model derived from classical-mechanics where we drop Galilean relativity and introduce the additional axiom of invariance of speed of light? In the latter case how can we be sure that this model does make sense? Isn't this hybrid model inherently inconsistent/incoherent? From inconsistent models we can derive any contradiction.
But the events A and B also correspond to positions A and B on the train. Let M' be the mid-point of the distance A —> B on the travelling train. Just when the flashes 1 of lightning occur, this point M' naturally coincides with the point M, but it moves towards the right in the diagram with the velocity v of the train.
- If we are assuming invariance of speed of light then "$M'$ moves towards the right" only within the embankment reference. From the train reference $M'$ would not move ($M$ would in the opposite direction). Why do we choose this particular reference and not any other inertial reference?
If an observer sitting in the position M’ in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning A and B would reach him simultaneously, i.e. they would meet just where he is situated. Now in reality (considered with reference to the railway embankment) he is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A.
- Wouldn't this description violate the assumption of invariance of the speed of light? It would amount to say - as far as I can see - that from the (inertial) reference of the train the light traveled equal distances with different times. Are we assuming invariance or not?
Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A.
- It seems to me that from the reference of the train the situation is completely symmetric (unless we are assuming there is a special reference like aether, are we?): they are two inertial references with a relative speed and the light sources have $0$ initial velocity for both (i.e. it moves at $v=c$ in both references), so both observer should have the same experience. Putting it in another way: we could just think that we have a stationary train and a moving "rest of the world" and then we could riproduce the same reasoning we did before arriving to the conclusion that the events are simultaneous for the observer on the "static" train and not simultaneous for the observer on the moving embankment. What is there that breaks the symmetry?
Some remarks after understanding better
There is not a "model" for the dynamics but there are some principles or axioms that we can imagine will be part of the RR model and which have a small intersection with classical mechanics so it is reasonable to expect they will not be inconsistent/contradictory
For some reason I have been thinking that the effective actual simultaneity of the bolts was something that was stated or implied in advance with the description of the setting, but Einstein ask us to stop thinking in terms of absolute time references. This is why I have been thinking that there is a symmetry in the descriprion of the events from the two references.
I think there is something slightly misleading in the final part of the argument when Einsten says:
Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A.
In fact the only concept we are considering here is a purely techical definition of "simultaneity" and the observer just notice that this definition doesn't apply in his reference, but this is not an actual simultaneity (or actual non-simultaneity) in the sense we are familiar with (with a reference to absolute time): it is just a completely new definition and we just discovered that it defines a property that depends on the reference.