I just have a hard time putting my thumb on one specific point where a person can consider themselves an alcoholic.
I've been a recovering alcoholic for going-on six years now and I can't tell you who is and who isn't an alcoholic either. There are definite warning signs because of the biological factors involved in alcoholism, but ultimately it's a decision that has to be made by the individual, or by people effected by the person's behavior.
So while there is no simple answer, these are the things that I would ask someone who asks me about it:
blackouts - a blackout is caused by the brain being overwhelmed by the processing of alcohol after it has been processed by the liver. In alcoholics, the liver converts alcohol into a more complex molecule than that of non-alcoholics, thus the brain has to exert more resources to dealing with it and starts to shut down other processes - one of which is the conversion of short-term memory to long-term. You don't have to be an alcoholic to reach this point, but if it doesn't take a lot of alcohol to reach it then is should be taken as a warning sign.
tolerance - the brain and body of an alcoholic will react in different ways to alcohol - either by increased or reduced tolerance to alcohol. So having a high or low tolerance to alcohol could be a warning sign. For me, my tolerance would fluctuate by weeks or months, meaning I would have high tolerance for a while then low tolerance, (not withstanding factors like food or body-weight). That told me that my body was reacting abnormally to alcohol, which was a sign that I was an alcoholic.
family history - every doctor will ask this question first. The scientific and medical communies have not questioned the genetic factors of alcoholism and addiction for decades. Specific genes have been identified, and every study in 50 years has shown the correlation between family history and alcoholism/addiction. Thus the practical application of this information is: "do you have a family history?"
cravings - alcoholics crave alcohol. You think about it when you shouldn't. You want to drink even when you've decided not to. The idea of quitting scares you. You can't stop drinking after two beers. All of these are manifestations of cravings that the brain creates. It's the same mechanism that causes us to crave salt, sugar, fat, sex, fight-or-flight reactions, etc. It's the reason why people continue to drink after it's become a destructive element in their lives.
These are the most obvious clues to determining alcoholism, and because only the alchoholic knows if he's blacked out, had cravings, or has high or low tolerance, only the individual can determine whether he/she is an alcoholic.
People will argue that it's all a matter of willpower and "weakness", but nobody who I've ever heard make that case has ever known the first thing about biology. Willpower and strength are what brings an alcoholic to actually acknowledge that they have a problem and quit, and if they can't quit on thier own, to get help.
I hope this helps.
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