Shakespeare considered himself as a poet.
During the time Shakespeare was writing, the terms "poet" and "playwright" were both used to refer to people who wrote plays. It is impossible to know what Shakespeare considered to be his profession. We know that Shakespeare acted in some of his plays. Perhaps he considered himself an actor? Unless someone uncovers a document written by the Bard himself that states what he thought he was, we will never know.
If you're interested in Shakespeare there is a really funny and informative book by Norrie Epstein called The Friendly Shakespeare that I would reccommend checking out.