Simultaneity on the train to Winchcombe

Sat in the steam train at Cheltenham I was reminded of Einstein's work on special relativity. He suggested that someone stood exactly in the middle of the carriage and had apparatus that was able to shine a light beam instantaneously in both directions. The doors at either end have light activated locks that open the doors when the light beam hits them. The person in the middle of the carriage sees each beam hit each door at the same time so both doors open at the same time. The light has traveled equal distances in equal times. Someone on the platform as the train goes by would see it differently. The back door would be moving forwards towards the backward travelling light beam. The front door would be moving away from the forward travelling light beam. So the person on the platform would see the back door open first and the front door second. Both witnesses of the same event but they see a different order of door opening.