[mnkv]

You understand some of what your cats mean because you learned it using the same language games Wittgenstein describes. Also they co-evolved to work with us. But just because you understand three moods of your cat, doesn't mean you would understand their language (if they had one). In fact, it is well studied that cats communicate differently with other cats than with humans.

Sidenote: grouping together non-verbal communication and language fails to take into account the richness of language.