The thing to remember about that "moving" observer is that in their frame, they are not moving.
Originally Posted by Bob Angstrom
Further, they are (as you note) in the middle - of where they consider the flashes to have occured.
So, if the flashes were simultaneous, in their frame, they should see the flashes at the same time.
(I think half the problem with the train-embankment experiment, is it's too hard to give up the idea that the embankment is "really" still and the train is "really" moving. That skews the thinking.
I suggest: convert the train to a rocket and the embankment to another rocket. Both observers (Alice and Bob) can see they are in relative motion, but they are in deep space with no external reference so can't determine which is "really" moving and which is "really" still. They both calculate physics and observe reality as though they are still. Alice thinks she's the one who's still and Bob is moving. Bob thinks he is still and Alice is moving.
Both consider the flashes to have occured at the ends of their own rocket. Both consider themselves as at rest. Both consider themselves as at the centre of the two flashes. In this case, simultaneous flashes should be seen by an observer at the same time. Since only one could actually see the flashes at the same time, they can't both consider the flashes to have been simultaneous.)
It's not just about seeing flashes at different times. An observer 1/3 of the distance between the two sources will see one flash twice as soon as the other, if they were simultaneous (in their frame). They would still consider the source events to have been simultaneous.
To write-off the experience of a "moving" observer not seeing the flashes at the same time as simply being a result of them no longer being in the middle of the sources, ignores that in their own frame they are not moving.
Get up, a get-get, get down.