I know similar questions have been asked here before, but none of them seem quite to address my particular confusion.
I'm not afraid of math (I did well in calc III last semester, for example) but my familiarity with physics is still primarily conceptual. My basic, high-level understanding of special relativity is that space/time must be malleable in order to preserve the constant speed of light regardless of reference frame.
Working from that understanding, the train & platform example where a flash of light originates from the center of the train makes sense to me. I think these two pictures (taken from Wikipedia) make it pretty clear that the two observers will disagree about whether the light reaches both ends of the train simultaneously:
However, I don't understand this example where lightning strikes both ends of the train simultaneously from the perspective of the stationary observer on the platform (also pictured on the previously linked Wikipedia page). It makes sense that the observer on the platform will see the front flash reach the passenger before the rear flash -- since the passenger is moving toward the front flash from the perspective of the observer on the platform -- but shouldn't the the passenger see the two flashes reach her location in the middle of the train simultaneously as well? Otherwise, how can we say that the speed of light is unaltered by the motion of the passenger's reference frame?
In other words: if moving toward a flash of light allows you to reach it sooner, wouldn't that contradict the fundamental premise that the speed of light is unaffected by relative motion?
I know I'm misunderstanding; I'm just not sure how to improve my understanding... Maybe someone can help me with the math, or at least explain what I'm missing conceptually?