Great explanation. Love the demonstrations.

You always seem to know what is puzzling me at that point and explain it perfectly!

"Observers in relative motion do not agree what happens at a given moment." This is only true if signalling via light is part of the observation process to get this effect. You can set up different schemas where everyone both on and off the train agrees. For example, supposing the presidents sit 10m from each other, set up two poles at a 10m distance from each other on the platform (assume the distance from the pole to the president is practically zero). When each president passes their respective pole they sign. Individuals situated on the platform next to the poles will register exactly the same time as individuals on the train who are standing next to the president.