|Well what can we say about this; one of the most discussed thought experiments conceived by Einstein; yet one of the most misunderstood pieces in all of Relativity.
When I first read chapter IX of Einstein’s wonderful exposition:
‘Relativity: The Special and General Theory. 1920.‘
I found it made good sense being straightforward and logical and well reasoned making it relatively (groan) easy to understand.
The great man put much thought and effort into describing and explaining this thought experiment, repeatedly emphasising that he was treating it from the perspective of the observer on the Embankment; which is what we would expect as it is all about Relativity.
The experiment can only be described relative to some particular reference frame; in this case it is that of the Embankment.
Chapter IX. The Relativity of Simultaneity
“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. People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every Event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises:
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. 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 (note1: As judged from the embankment.) of lightning occur, this point M’ naturally coincides with the point M, but it moves toward the right in the diagram with the velocity v of the train.[Fig. 2.] 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.[Fig. 3.] Now in reality (considered with reference to the railway embankment) [Fig. 2.] he is hastening toward the beam of light coming from B,  whilst he is riding on ahead of the beam of light coming from A. 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. We thus arrive at the important result:
Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an Event.”
The reference numbers inserted into Einstein’s writing match those in the diagram, Fig. 2. This is Einstein’s Fig. 1. expanded to a Space-Time diagram with the Frame of Reference of the Embankment shewn in blue with the train in red.
Einstein asks the question “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?”
So what exactly is he asking here? It seems to be a straightforward query about how the two lightning strikes are related when considered relative to the train; yet we are dealing with two distinct reference frames here: that of the Embankment and that of the train and the answer will be different for each.
First of all Einstein introduces us to the perspective from the Embankment and only from the Embankment. That this is indeed the case is quite clear when considering his argument point-by-point:
What else can vice versa mean but that the whole of what has been written above would apply in reverse if considered from the train observer’s point of view in position M’? – “The Relativity of Simultaneity” – then surely it means: “That simultaneity is only relative to the reference frame in which it is measured”.
In Fig. 3. the same events are drawn using the Train’s Frame of Reference. The Train is of course now stationary while the Embankment is moving with the velocity -v. The reflected lights meet at . M’ meets the light from A  first at time tA , then the light from B  later at time tB .
(After some debate I have retained the convention of making the moving frame the primed frame, so M’ is mid-point between A and B on the Embankment as it is the Embankment that is moving in the Train’s Frame of Reference.
For the observer, sitting in the middle of the train, his frame is at rest and point M, mid-way between A and B, is permanently located in his Frame of Reference; and the embankment would have mid-point M’, moving toward strike A. Leading to the inevitable conclusion that the observer at M in the train would see the strikes simultaneously, while the Embankment observer at M’ would see strike A before strike B.
The constant events that will be the same from whichever Frame they are observed are the two pulses of light meeting at the midpoint of the line AB. The question we must ask is: what are the coordinates of that location in the individual Frames of Reference?
Now within each Frame of Reference we have to consider what is at that mid-point of AB at the time the pulses of light meet. And that this could be either M or M’ depending on which frame is stationary for the particular observer. M for the embankment and M’ for the train.
That is the Relativity of Simultaneity; each observer will see it within his own Frame of Reference but deny that it happens in any Frame of Reference that he observes to be moving.
Which is exactly what Einstein was saying when he wrote:
“People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every Event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment [my emphasis].”
How much clearer and more explicit can he possibly have stated it? Repeating the principle in his conclusion where he states:
“Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity).”