Sadly, it's this whole "publishing thing" that leads writers to doubt that they are -- writers. Society has made us believe that unless we're published, we're not validated. That we're not really writers. Hogwash.
Allow me to illustrate: Let's pretend that next month, I will finally nab that long-sought-after agent (ha!). Well, the book that I've spent hundreds of hours creating, editing, polishing, in the past year and a half is THE book the agent will take on and try to sell to publishers, right? And if she does sell it, then bam! Does that suddenly make me more of a writer than I was a few weeks before, when I was finishing the novel (and didn't have an agent)? No! In fact, the material itself hasn't changed a bit - only the results.
I heartily reject the idea that one has to be "published" in order to be a writer. Because you can't GET published without having written something. A writer WRITES.
So, today, if you've written anything creative, give yourself permission to call yourself a writer. Be comfortable with that title. Try it on for size. Because you've earned it. And no one can take it away.
By the way, here's a great article on this topic, from the GLA blog - At What Point Can You Call Yourself a Writer?