Third person is a imaginary viewer/camera following a character around. The POV is that. FP is looking at the world through the character's eyes.
No, third person in this game is simply a representation of a FP perspective. The fact that you look at your character from a third point of view doesn't mean you aren't that character, it just mean that for gameplay dynamics inherent to the game the developers decided that a third view point was better to control the character and the context of the game itself.
As an aside note when you remember an event you usually recollect what happened in a third person perspective and not in first person (also if you don't see yourself the POV is not usually in first person). The same happens very often in dreams if you can recollect what it happens there in details. It is very rare for the imagination to recompose a scenario from a first person perspective, you have to somewhat "force" it.
But that's circular reasoning. I am arguing they should not, not that they do not.
Both the horse effects and the mud, etc. should not be visible in Third Person. Because are things that are viewed from Geralt's perspective (on his face, via his eyes,etc.) but somehow applied on the imaginary camera following him.
It's not circular reasoning. It is just how the devs developed the game. You can also not agree with that or not like it, but that's how the game plays and hence it makes complete sense for an horse trotting to do a step/pause/step because that's what it feels like when you trot an horse.
One of the most important key components of a good gameplay is consistency. No matter the vision you decide to go with the most important thing is that you must stick to it, elsewhere nothing will make sense in the end.