I would say the distinctions are all very fuzzy, particularly between astronomy and astrophysics. One might be more inclined to associate observational topics with astronomy and theoretical ones with astrophysics, but that's not always how the words are used in practice. Cosmology is a little bit more distinct. It refers to topics that impact the universe at large (or, at least, the universe as we presently know it). However, most astronomical topics have cosmological relevance, so even this can be a bit ambiguous at times.
In general, I don't shuffle posts between astro forums unless a topic is very clearly cosmological (like Hubble's law or WMAP) or not cosmological (like extrasolar planets).