You can drink unhealthy amounts of alcohol without being an alcoholic.
And all "drinking in the morning" (when it happens on a regular basis) means is that the withdrawals (which I had on a daily basis when I quit, and which did lead to drinking on many mornings) are uncomfortable enough that it takes a drink to feel "normal". You can be an alcoholic without ever drinking in the morning, though, and some people occasionally have a morning drink and it doesn't mean a thing except that they felt like having the bloody mary or whatever.
It's all about what alcohol does for you, and how important it is in your life. Once it becomes something you are mentally obsessed with, that's a pretty sure sign.