Jason, I agree with you that there is a difference between writer and author. I also agree that in order to call yourself an author, you have to have published a book — fiction or non-fiction.
But, and this is just personal preference, I will always refer to myself as a writer.
I can’t recall where I heard/read this but someone said the difference between a writer and an author is this:
A writer is a person who writes.
An author is a person who has written.
A writer is immediately on to the next story, the next work — they are always writing. The work never stops.
An author is someone who spends time promoting their last story. Their latest work or masterpiece must be promoted and discussed to ensure that all of their work isn’t wasted.
Does it matter? Not really. Only to the individual. And for me its a mentality thing or a philosophy for how I want to approach what I do.
I’m working on a novel. I’ve written several (dog shit) screenplays. I write essays, material for Medium, my full time job is as a Technical Writer for a software company, and I do some free-lance work providing content for blogs.
So I prefer to go by writer. Even when (if) I finally finish and publish my novel, I will continue to call myself a writer.
Again, this is all virtually meaningless in the grand scheme of things, but I think it ultimately comes down to individual philosophy or world view.