I have a pet peeve as a writer. I hate it when people say things like, “You’ll always be a writer whether anyone reads your work or not,” as if that was supposed to be comforting in some way. It has always struck me as being a bit like saying, “Don’t worry Sisyphus, you’ll always be a boulder pusher, whether you ever get to the top of the mountain or not.” That’s more of a curse than a comfort.
Of course it matters to aspiring writers if their work is published and people read it. Of course it mattes.
“Aspiring” is a troubling word in itself, really. It irks writers with way it lumps in rank amateurs with those who have labored long and hard with serious intentions. (Sisyphus was an aspiring boulder pusher the first time he headed up and is still an aspiring boulder pusher on his 200th try.)
Beyond that, unpublished writers (or writers who have published some things but not their Great Works) are not aspiring to write. They are writers. That is of course, what those well-intentioned friends are saying with their “you’ll always be a writer, it’s just who you are” pep talks.
The language seems to be lacking a word for what such writers aspire to be. “Published writers” is closer, but it isn’t entirely it. There are few writers more miserable than those who have published books that do not sell. “Read writers” is more like it, but it is a terrible phrase.
What I think it boils down to is that the artist is not seeking an identity or a title as much as a relationship. The goal is to connect to other human beings by sharing the work. In that sense saying “of course you’re a writer” is a bit like saying “Don’t worry, you’ll always be a woman whether or not you ever have a boyfriend or husband.” This may be true, but it is not much of a comfort to someone who would like to find someone to share her life with.
In conclusion– yeah, I haven’t really got a conclusion, except to say maybe our language needs some new terminology to express the artist who has completed the artistic process by forming a relationship with an audience. Not just a writer but a person whose writing is read.
Story and Self 
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