You can be addicted and have a drinking problem without being an alcoholic. In my book, an alcoholic is someone who is physically-dependent, gets withdrawal symptoms when stopping, and engages in a practice called 'relief drinking'; drinking specifically to ward off withdrawal symptoms. What I disagree with is the use of the term for less serious degree of problem-drinking. I disagree with the binary view that one is either an alcoholic or not. Lots of people feel the same way.
I have to say, as a newbie, I found this comment slightly triggering. Completely my problem, not yours! But my first thought was, "Ooh, so this guy is saying that because I didn't get the shakes or anything when I quit, that means I'm not an alcoholic. Maybe I could get away with having a drink or two after all."
If I'm sensible and honest with myself, I can recognise that it doesn't matter what word/noun/label you use - if you can't stop after one; if you get anxious whenever you're forced to spend the night without alcohol; if you promise tonight will be the last time, then it isn't....then you have an issue that likely needs attention.
NB: According to the dictionary, alcoholism is the continued excessive or compulsive use of alcoholic drinks. It's a pretty broad term...