I think the moment you give life in a physical form to your thoughts, you have WRITTEN and you are a WRITER, no matter how, where and what you have used to do so.
Does the act of writing "I'm hungry" on a piece of paper/Word document/etc. make me a writer? In the technical meaning of the word, yes. A writer is the one who writes. Maybe we should be simply called authors, as ones that "originate or create", according to the dictionary definition of the word.