True, BUT, if you were pushing the wrong "buttons" consistently, I think you would be UNtraining the horse. You are reinforcing the proper training, so you ARE training it.
You would be training the PERSON, not the horse...the horse already knows, the person doesn't. Take my horse for instance...you could push the wrong buttons....but there is no way you are untraining him.
I'm also a very strong believer in that a horse who's not entirely honest (yes there are bombproof babysitters who make all other horses look bad) will act and work differently due to the rider's attitude, ability and leadership. These horses may be the one's who seem to be "untrained" because they are so willing to take the easy way out for themselves??
Back on topic of the thread though, I agree that there are a million ways to train a horse and there happen to be a million trainers out there. Everyone has a way that seems to work best for them or that they are most comfortable with. The good trainers, IMO, will keep an open mind when their ways aren't successful and try other things or admit that the problem is something they are unsure of fixing with that horse.