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The point of the paradox is precisely that macroscopic does not seem to imply classical, contrary to our intuition that objects we see with our own eyes naively seem to always be in states of definite position and momentum instead of states actually allowed by quantum mechanics.

A "macroscopic" object is simply one that consists of many atoms and exists at our usual scales. A cat is clearly macroscopic, and in our everyday experience where it is not tormented by mad scientists it can be well-described by classical mechanics. So the crux of the paradox is simply that it creates a situation where the classical description of the cat does not apply, which you are supposed to find counterintuitive, if not impossible. If you don't, then there is no paradox.

There cannot be a proper definition of "classical object" because nothing is classical. It is generally believed that quantum mechanics is the proper description of the world and that everything is ultimately a quantum object - "classicla objects" are simply those where we can get away with describing them by classical physics and still get the relevant points right. What the "relevant points" are changes from application to application, so there's no unique idea of what objects qualify as "classical" - it depends on the specific situation and your goal in describing it physically.

In particular your idea that a classical state could somehow be a state in Hilbert space is not correct. Classical states are states in a classical theory, not in a quantum theory, there is no Hilbert space in a classical theory unless you insist on the awkward Koopman-von Neumann formulation. You need to take a proper classical limit of the quantum theory (heuristically, $\hbar\to 0$) and see which classical states the quantum states are mapped to, but this will highly depend on the system and the precise general procedure for this classical limit is, to my knowledge, unknown, though actively researched.