A drinking problem is one where drinking contributes to you not fulfilling your responsibilities. For instance, if your job performance decreases, you get into some kind of legal trouble, or you don't pay your bills.
I've used this measure to check my drinking habits.
What does it mean to be an alcoholic in this case? That Wikipedia page doesn't give a definition, nor does the linked NIH press release.
The college alcohol study from Harvard lists criteria, but they are all generally things that have clear negative impact. i.e. alcohol-related school problems, legal, interersonal problems. These are things that fit the traditional definition of alcoholism, i.e., "your drinking is screwing up your life". The college alcohol study also makes no mention of "high-functioning alcoholism" in either the press release or abstract, so I don't see how it's relevant to the article at all except that someone thought it lended credence to the claims.
The "high-functioning alcoholic" categorization seems really fishy to me. Somehow a person is overachieving, has no negative effects from alcohol consumption, and yet is an alcoholic? On what grounds is that claim made? Because they drink more than someone else thinks they should?