I think it depends a lot on the nature of your world. In most of our myths, monsters are a product of some breach of the ‘natural order’ for instance, I don’t believe a centaur is ever described as a monster, though they are often depicted as a menace. The Minotaur on the other hand was a punishment devised by the gods, and I think could be considered a monster.

If you don’t want this system you could simply state the monsters are an especially dangerous type of animal, and could include, say, a specific dangerous wolf as a warg while generally excluding wolves that keep to themselves.

What is and what isn’t a monster can be (probably should be) culturally defined unless you really need something to only interact with monsters, like Pokémon, or D&D’s distinction between charm person and charm monster.

I believe there is no strict rule for how to distinguish between monsters and animals, beyond those you already suggested. While you can pick freely which characteristics distinguish them, I’d suggest tailoring them to your world. They should mean something. Perhaps monsters are magical to show off the magical system/ideas of your world. Perhaps they are tormented souls of humans to show how grim your setting is. You might make monsters more intelligent to help people emphasizing with them, or blur the line completely by just making monsters a fictional kind of animal to support a scientific narrative.