If you write every day and read every day, then you can consider yourself a writer. If you document your reading, viewing, and listening experiences, then you can consider yourself an active writer. If you extract from those experiences ideas for your next essay, story, play, or poem, then you can consider yourself a creative writer. If you receive on average a dozen rejections a week from publishers and simply file them away just as you would your far fewer acceptances, then you can consider yourself a dedicated, no-nonsense, self-possessed writer.
Writer Jeff Goins has something useful to say about the writing life in his highly recommended blog post "Everything I've Learned from 6 Years of Writing for a Living." It's a worthwhile read if you are considering a writing life, writing what you believe to be your magnum opus, or receiving countless no thank-yous for your submissions.