I hope none of us are deluded enough to think the first definition is appropriate - that horses feel a deep admiration for us based on our abilities, qualities or achievements...
The second, a due regard for the feelings, wishes or rights of others seems to be putting a lot on the horse!
I've met a number of horses who apparently 'respect' their owners who are simply scared of them - the horse won't barge into the person, or turn his hind quarters to them, because they fear the consequences of doing so.
And 'disrespect' often amounts to fear or poor training. The horse who turns her back on you in the stable may simply be retreating because she wants to avoid you, or what you are about to do - for example, maybe her saddle doesn't fit and she is trying to avoid it. The horse who pushes past you in the field may be retreating from a threat made by another horse, who is likely to kick him much harder than you will if he doesn't get out of the way! And sometimes, the horse who 'invades your space' has simply learned that it benefits him in some way to do so - people back off when he does this, for example. That is the trainer's problem, not the horses, although the horse usually gets the blame.
The words we use to when we talk about our horses really matter, and this is one that seems to me to place blame on a horse when, in most cases, he is simply trying to tell you he has a problem.