The set-up. The platform is a distance L long and its observer M stands in the middle. Two lightning bolts strike the two ends at the same instant, in the platform frame. A passenger P sits in the middle of the train. P passes M at the instant of the strikes.

The platform frame. The flashes travel the same distance L/2 to M at speed c, so they arrive together. P is moving towards the front flash and away from the rear flash, so P meets the front flash first: t_front = L/2(c + v) and t_rear = L/2(c − v). P’s own stopwatch runs slow, so it shows these times divided by γ.

The train frame. P is at rest, in the middle of the train, and light travels at c both ways. P receives the front flash first, but the two strikes were the same distance away. So in the train frame the front strike must have happened first. The time between the strikes is Δt′ = γvL/c² (from the Lorentz transformation, Δt′ = γ(Δt − vΔx/c²) with Δt = 0 and Δx = L), and it equals the gap on P’s stopwatch.

Who is right? Both. Events at different places that are simultaneous in one frame are not simultaneous in another. But the order of the two flash arrivals at P does not depend on the observer, because the flashes reach P at the same place. Only the order of the strikes depends on the frame.

The train in the picture is as long as the platform when at rest, so each point of view sees the other contracted by γ.