The “are you a real writer” debate is one I’ve witnessed more times that I can count in my sixteen or so years in this business. For the most part, I think it’s people letting their egos talk a little too loudly.
If you write every day, and what you write is unpublishable, are you a real writer?
If you write a couple days each month, and what you write is pure gold, are you a real writer?
If you write once a year, and never show anyone what you’ve done, are you a real writer?
Does it even matter? If you sit down with the intention of writing, doesn’t that make you someone who writes, ie, a writer?
What’s the point in trying to define what a “real” writer is other than stroking your own ego (“I write *every* day!”; “I write when the spirit moves me and what I write are beautifully crafted tales!”) while getting into an Internet pissing match, to be blunt about.
I think the better question is: as a real writer, wouldn’t you be better off spending your time and energy on your new book instead of debating an existential question on a message board? 
I love to write. I write regularly and that makes me a writer in my own mind. Debating whether a person is a writer or not is definitely silly and a waste of precious time.
I consider myself a writer because now I’m writing with the goal of publication and continually work to improve my craft. I think it’s essentially all about passion. If writing is your passion, then you ARE a writer.
I think anyone who writes is a writer whether it is shared, blogged about or kept under lock and key in a journal.