I think it was implied. I mean you wouldn't call a guy who can make a Grilled Cheese a chef would you?
No, but I would call that person a cook, which is a more analagous word. Anyone who writes is a writer. Anyone who cooks is a cook. Nothing about ability is implied by either noun.
I'm not so sure though... my boss writes me e-mails about projects I'm working on in my company, but my boss makes no claims of being a writer. Every child in school is required to write term papers, but are they all writers?
^ Yes. The word writer makes no assumption about quality or intention, only the fact of the person engaging in the act itself.
I've decided to keep any last names out of the book for now, as my book does have a unique naming convention. If the publisher thinks last name would be better that would be easy to add in the future.
The Name of the Wind is book chalk full of characters who all lack last names. Though the importance of their first names kind of makes up for it.
I think it would depend on semantics... Without copying the link or quoting the (copyrighted?) definitions, according to Random House dictionary, def. #1 refers to someone who writes books, etc. as an occupation or profession; an author. def #3 is what you are describing--anyone who writes something. So really, it all depends on what definition of "writer" one uses.