Absolutely it would make him an alcoholic. Alcoholic would be defined as someone who HAS to have alcohol. You dont have to drink a lot to be an alcoholic, you merely have to be someone who HAS to drink.
He obviously HAD to drink, as when he tried not to, he couldnt do it. You say you have a glass of wine, thats fair enough..but ask yourself, could you not have that wine, or is it a case that you MUST have that wine. That answers the question of addiction to alcohol.
MMO addiction is much the same, the key is between HAVING to play and not having the play. If its a must play, a requirement in a persons life that they must play game X, that they cannot go without playing game X. Then by its very definition, thats an addiction.