It is obvious they use several different horses as Rusty. In 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.
When the kids cannot get the vehicle started, the father realises that the spark plug wires have been pulled out. However, when the vehicle starts, it is a diesel engine running, which does not have spark plugs.