You are a writer whether or not anyone has read your work yet. You are a writer whether or not you have been published, or won awards, or got good reviews. There is never any badge or certificate you'll get that will say you are a writer. You just have to believe it and it's true.
If you believe it for long enough, and finish enough stuff, then you will gradually transform from being just a writer into being a good writer. The publication, awards and good reviews will follow - in direct proportion to the strength of your belief.
(If you can't bring yourself to believe you're a writer, pretend. Act as if you are and, actually, that's all it takes. Acting like a writer makes you, in fact, a writer.)
There are many different kinds of good writers. Some are popular, some are deeply insightful, some are just good for passing the time in an airport lounge.
The difference between a good writer and a bad writer is that a good writer finishes things and moves on.
If you've been working on an idea for the last five years to the exclusion of all else, or you've started ten novels but they all trailed off somewhere around chapter thirteen, then you are a bad writer. It's not about the quality of your prose, it's about what you do with it.
A good writer knows that ideas don't fail. If he's writing something that doesn't feel right, he finishes it anyway, and moves on.
A good writer has faith that more good ideas will come.
A good writer produces enough material that he's not afraid to shelve the crap.
A good writer says, "I am a writer," without qualifying the statement.
A good writer accepts praise and rejects criticism.
At least, all that's what I keep telling myself. I'm still working on being a good writer.