Meno,
“First let’s take a simple case: Light travels 186,282 miles per second. Let’s say I’m running in a straight line at 1000 miles per second, and lightning strikes exactly 186,282 miles ahead of me. How much time passes (on my watch) before I see the lightning? The answer, I believe, is exactly 1 second, because light travels the same speed in every reference frame – that’s Einstein’s chiddush.”
Correct.
“So now the train example. To make it simpler, lets say the train is 372,564 miles long (2 x 186,282 miles). The guy on the platform sees both lightning strikes simultaneously after 1 second – simple. But why doesn’t the guy on the train also see them simultaneously? Why is it different from my case? Is my conclusion in my case incorrect?”
The flashes were 186,282 miles from each observer, so they both figure that the light from each flash took 1 second to reach them. But the observer on the platform saw the flashes as happening at the same time, and the one on the train saw the flash at the head of the train happen first. In the inertial frame of the platform observer, the man on the train was moving away from the light coming from the back flash and towards the light of the front flash, so the light from the front flash traveled less than 186,282 miles before reaching him, and the light from the back flash traveled further than 186,282. But to the man on the train, the light from each flash traveled exactly 186,282 miles before reaching him.