Major goof
It is very obvious they use several different horses as Rusty. Most scenes Patrick is riding a horse with 3 stockings and 1 left back fetlock (common horse markings) and in other scenes such as the one where he is showing his family his jumping he is riding a bay, and then switches to a horse with two socks.
I have been riding horses since I was a little kid and I can tell the difference and I'm sure anyone who takes care of horses can too.