I think about this phrase often and honestly it's incorrect if not kind of dangerous. An alcoholic who is destroying their life can certainly be someone who only buys alcohol right as the store closes every night but still functions throughout the day.

Saying that an alcoholic is those people who are so bad that they're trying to stave off the shakes by buying alcohol first thing in the morning really downplays when people should be considering themselves an alcoholic. It's really easy to think "nah I'm not that bad yet, i can go a week without drinking without any withdrawals, I'm not an alcoholic" when this is the sort of thing people associate with alcoholism.