Ours have seldom been shy about peeing. Not sure if they learn by watching the older horses. They pee under saddle, in harness, in the barn aisle after a long drive and in the trailer. When we drove more miles to compete, they were “truck trained!” Riding in the stock trailer they would pee when you turned off the truck engine. In the semi, they would pee when you put on the air brakes! They knew they would not be moving for a while, best make use of the opportunity!!
In use, we give them opportunities to pee by stopping on a grassy road edge, standing for a while. Most effective with mares in season who need to go more often. Riding is similar, stop by the road edge on grass and wait a bit. Usually get a result shortly.
One older, experienced horse would pee when loaded in the trailer at shows. He would not let his guard down with other horses moving around him. We figured he felt protected with trailer walls or stall walls around him so he could pee safely! 99% of the time he would pee in less than 10 minutes, took him back out for more classes. Told daughter he could not pay attention when he has to pee! She TOTALLY undestood!! Ha ha
I would rather clean trailer twice than have a horse who won’t go.