Muslim = shorthand for practising Muslim. You are right, some Muslims do drink. And some of them end up in AA too. Ditto Methodists. Ditto (especially in America) teenagers and even children.
I was in no way suggesting that your clinical depression was why you were on the pity pot.
I was talking about this:
"I feel like scum. So many people reached out to me at AA & here I am, drinking....."
and this:
"So thats where I'm at. A silly, weak willed woman"
You've been to AA. You will have heard people say to you, "keep coming back." Now, if you don't want to go back, then of course you shouldn't, but if you do want to go back - because people reached out to you, because you were happy there, because you will miss the meetings - then what are you waiting for? For people to phone you and beg you? Go to a meeting if you think it will make you happy and worry later about the textbook definition of an alcoholic.
Why do you feel like scum? because you've had a drink? Normal people drink. And alcoholics who don't treat their alcoholism drink. So whether you're an alcoholic or not, you've done exactly what society expects of you. How does that make you scum?
As for willpower - again, it doesn't come into it. You might as well try exercising willpower on diarrhoea. Alcoholism is an illness and you don't think yourself well.
Or, again, the alternative is that you're not an alcoholic, in which case you don't need willpower not to drink, you just make a choice and it happens.
Basically, either you are an alcoholic or you're not. If you are, it doesn't make you a bad person. But nor can you choose not to be. If you are, you need to get help from somewhere or it will kill you.
I would therefore suggest that, until you know for sure one way or the other, you assume you are one and act accordingly.
Hope you find this more positive.