I don't entirely agree, but stress that my answer is only my personal opinion.
I think that if I am speaking about the mouth, nose/muzzle and eyes of an animal, I use 'face'. If I don't consider the eyes, then I use 'nose' if it is fairly short, or the animal is my pet, and 'muzzle if it's fairly long and I don't think of the animal as friendly.
My view on this is that primates would have a face, since they are much like humans. However, non-primates would not have a face. So you could say, for example: "The gorilla turned its face towards me." But you would not say: "The dog turned its face towards me." If you wanted to express this idea, you'd say something like: "The dog turned and looked at me."
To me, they all have faces. (Even a clock has a face.) Back to the dictionary, which probably shows why opinions differ:
Face refers to the front of the (usually human) head (the surface of the front of the head from the top of the forehead to the base of the chin and from ear to ear.
I have no problem at all saying "That corgi has a very sweet face" or "Pugs always look worried, but I guess that's just the kind of face they have." To me, these are all equally valid: The gorilla/dog/mouse turned its face towards me. The gorilla/dog/mouse turned towards me. The gorilla/dog/mouse turned and faced me. The gorilla/dog/mouse turned and looked at me.