By way of Dave White's Writing Block:
What is a writer?
A writer is engaged in putting words on paper as a means of discovery such that if he hasn't done it in a while, if he hasn't ventured something new, he isn't a writer.
Who is a writer?
Anyone who acts on the compulsion, then freely stays in the habit of writing, honing the ability to communicate.
Are you a writer?
I like to think so.