There are many different kinds of writer. To me, anyone who writes on a regular basis can call themselves a writer. The challenge is figuring out what subset of "writer" the person belongs to. For example, a copywriter probably wouldn't do a good job if they tried to fill in for a technical writer, and writing fiction is a different skill again.
A writer is someone who can effectively and ethically translate an end line (what you want to say) to a copy line (how will you say it).