Some professions have accrediting bodies that are responsible for designating someone as a "professional <whatever>" in that field for legal purposes. For example, I have an electrical engineering degree, I have a career in engineering, and it would be correct to refer to me as an engineering professional, yet I am not a "professional engineer" because I have not taken the professional engineering exam and been licensed as one. In fact, most engineers aren't professional engineers since it is only useful in a fraction of engineering jobs. Medical and legal professions have similar criteria and licenses.
Pool, on the other hand, does not. I can't really think of a situation where it would be important (in a legal sense) to designate someone as a professional pool player, so the only reason to start doing so would be as a business convenience, as in golf. There also doesn't seem to be a high enough demand for such a system to pay for the effort involved in maintaining it and doing whatever it would take to gain widespread acceptance, so I don't expect that to change.
Ultimately, that leaves it up to the pool community to decide what we mean when we call someone a professional pool player. In my opinion, a professional is someone who plays pool with the intent of earning their living doing it. If I want to talk about how well someone plays the game, I talk about skill levels or ranking. Naturally, there is a correlation between the two, so when someone talks about "playing like a pro", the implication is that they play very well. I think there is also an general understanding that when we talk about a "semi-pro", we mean an amateur that can (and does) compete in events that are attended by most of the pros.
In short, I think that playing a lights-out game of pool might be a prerequisite to being a successful pro, but it doesn't make you one.