To me, there are distinct differences between these three titles, and when I will allow myself to use them.
To become an author, you must sell something you’ve written. If you’re a traditionally published author, that means selling a book to a publisher. For independently published (read: self-pub) authors like I plan to be, it’s the first time you sell a book to someone who isn’t a friend or family.
If I’ve never spoken to you in person and you are the first one to buy my book, you will be the one that makes me an author. Until then, I am not. Aspiring author is the most I will claim.
As for becoming a full-time author, that’s the holy grail. That says not only is someone buying your work, you’re making enough money to support yourself with your writing, even if you still don’t make much. That’s the dream, because it means I’ll be able to write even more.
But writers? Writers write. If you write, you’re a writer. Maybe not a good one, maybe not a successful one, maybe not a widely read or beloved or influential one. Maybe you’ll never become any of those things. That doesn’t matter. If you write, if it’s something you love and seek out, you’re a writer.
Writers write. If you want to be a writer, write. That’s it.