Trying to understand one of Einstein's thought experiments I try to understand Einstein's
Relativity: The Special and the General Theory,
chapter IX., "The Relativity of Simultaneity". Here's an online version: http://www.bartleby.com/173/9.html.
             |----M'----v->
####-----A--------M--------B-----####

Einstein considers two lightning strokes happening at two points A and B
along a railway embankment. At this moment, two observers M and M' meet each other at the mid-point between A and B. M resides on the embankment, M' is riding on the train moving with velocity v towards B.
Then, paragraph 3 reads:

"... he [the observer at M'] is hastening towards 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."

As far as I understood SR until now, this is actually the non-relativistic
point of view. Bearing relativity in mind, the moving observer M' should
experience both light beams from A and from B approaching with light speed
as well, the beam from B being blue-shifted and the one from A red-shifted.
In respect to M' they have different energies, but they should arrive at
the same time.
The longer I think about it, the more I doubt whether Einstein found a
very fortunate thought experiment here. What do you think about it,
could that be true?
 A: This thought experiment is not only fortunately chosen, but crucial to understanding the implications of relativity on simultaneity.  It is not described in terms of the non-relativistic view. 
Observer M' observes the following:  1) The speed of light is c, independent of direction (hence he is viewing the world relativistically).  2) The lightning strikes leave marks on his train that are equidistant from him, so he knows that A and B occurred equally far away from him.  3) Light from B reaches him before light from A.  From these three observations he concludes (correctly for someone in his reference frame) that B happened before A.
Meanwhile, by similar reasoning, Observer M concludes (correctly for someone standing on the embankment) that A and B happened at the same time.
Since the observers draw different conclusions about whether A and B occurred simultaneously, the implication is that simultaneity is relative, i.e. dependent on your frame of reference.
A: "As far as I understood SR until now, this is actually the non-relativistic point of view. Bearing relativity in mind, the moving observer M' should experience both light beams from A and from B approaching with light speed as well, the beam from B being blue-shifted and the one from A red-shifted. In respect to M' they have different energies, but they should arrive at the same time."
In the paragraph you quote, he is analyzing things from the point of view of the rest frame of observer M, in which the flashes were simultaneous (this is just a physical assumption about the scenario being considered, of course you could also imagine a different physical scenario in which two flashes happened that weren't simultaneous from the perspective of an observer on the ground, but in this particular scenario they were). Based on this, he deduces that the signals will reach M' at different moments according to the clock of M' (note that predicting when light will strike a physical clock according to that clock's own reading does not require you to analyze things from the rest frame of the clock, you can use the time dilation equation to predict this using a frame where the clock is in motion). A key point to understand here is that all frames must agree in their predictions about local events, like what times two clocks read at the moment they pass right next to one another, or what time a given clock reads when the light signal from a distant event reaches it. If this wasn't the case, then different reference frames would be more like parallel universes that would predict totally different events. For example, say the clock of M' was equipped with light sensors connected to a bomb, and that the bomb was programmed to explode if the sensors received strong light within some very short time interval according to the clock. If different frames didn't agree about local events, they could in this case disagree about whether the bomb exploded, and about whether the observer standing next to it was alive or dead! Different reference frames are just intended to be different ways of assigning position and time coordinates to the same set of events, nothing more.
So, with that in mind, we conclude that all frames must agree the light from the flashes reached M' at different moments, including the rest frame of M'. But how can this be compatible with the fact that, according to the fundamental postulates of special relativity, both flashes must also travel at the same speed c in the rest frame of M' (something that would not be true from a non-relativistic point of view, assuming they traveled at the same speed in the frame of M), and both happened at next to ends of a train that are equidistant from M' in this frame? There is only one possible way of making this consistent--in the rest frame of M', the flashes must have happened at different times (i.e. they are assigned different time-coordinates in this frame). There is nothing problematic about the idea that photons which were emitted at different times will reach an observer at different times, even if both signals were emitted at the same distance from the observer, and both traveled at the same speed relative to the observer.
