I would say no, you aren't an alcoholic, because you have the objectivity to actually ask yourself that question. You are concerned enough about your behavior to reflect on it rationally. And I say that as someone who flirted with alcohol dependency about 10 years ago.

I didn't reach for a drink when I woke up, nor did I keep a bottle in the desk drawer at work. But I was miserable at work and started off by drinking a glass of wine when I got home, "just to take the edge off". That glass became two, then three, then the whole bottle eventually. My wife helped me realized I was drinking too much, and for the wrong reasons, and encouraged me to make the changes I need to make in my life so that I didn't have to compensate for it with alcohol.

Its 10 years later and I'm still very happy to drink socially at a party or have a nice bottle of wine with a nice meal. But that's it. Its a huge step, being able to ask yourself that question. I think the next question you need to ask yourself why you crave the drink, and if there is anything you need to change in your life to remove the need.