Honestly, I don't think there is much disagreement between anyone here, I believe it's just how people have said it. No one gives their horses choices, at least I really hope not. Obviously in some cases the horse gets a choice, such as if they are hurt, I hope to god we don't choose to ride them. But let's say out on the trail, if your horse wants to go left, and you want to go right, are you really going to let your horse choose which way to go? Hell, I let my horse choose and then tell her we're going the other way.
And from what I've seen with NH, the point is to make yourself their leader in a natural way. This may not be the case with all trainers, but from what I've experienced that's what they're trying to teach you to do.