Wattpad, Blogger, Wordpress, Medium, etc, are all valid and great options for self-publishing. Just because there is no physical copy in your hands does not make you any less of a writer. A writer is a writer. Period. Whether you write fanfics or short stories or book reviews or food blogs, you are a writer. By definition, a writer is someone who writes. Don't let anyone tell you that you aren't a "real" writer. The same goes for if you are self-published on Amazon KDP or Ingram Spark or any other self-publishing platform. You are an author. Just because you aren't traditionally published, does not make you any less of an author. Ignore the haters. Critics and cynics will always be jealous because you have done what they couldn't do. Do not let others define you. You define yourself. As cheesy as it sounds, it's true.
"Anyone can write a book" or "anyone can have a blog". So what? Just because anyone can write, it doesn't make it less valuable. Everyone can breathe, right? Does that make it less valuable? (Hint: the answer is no) The beauty is that everyone can write. We can see people's ideas in different mediums, whether that is writing, art, or music. Just like digital painting and acrylic, there are online published works and people that print hard copies of their writing.
Fanfiction is a great place to get started as a writer since you do not have to go through world building and creating characters. (As long as you don't claim the original ideas/characters as yours) But I do suggest that you move on from this if you want to write your own original stories one day. You're a writer if you write fanfiction but not an author since it's not your own idea. Building your own world is incredibly satisfying and your characters live in your mind. It is unlike anything else.
Wattpad is great for posting original works. So is Tumblr, Medium, Wix, Blogger, and many, many other platforms. You're a writer and an author if you publish original works online. Add it to your resume and tell your family! It's awesome! Even if your work doesn't gain traction, don't give up. It's still an amazing accomplishment and it's a good gift for friends and to pass down to family and for yourself.
Online works, whether it's food blogging, magazine writing, book reviews, or
anything else, are an great way to write and be creative. You don't have to
write whole novels to be called an author or writer. Being self-published is
still being published. Traditional is nice and I'm not knocking it, but
indie-authors still seem to not get enough credit. Writing, like any skill,
takes time to develop and in two years you might cringe when you look back at
old writing and that's fine. It shows how far you've come and by publishing
online your fans will have more appreciation for how you've grown. Publishing
online gives feedback and people across the world can have instant access to
your writing. It's a good idea to write as a hobby or in a diary or bullet journal
or wherever. So, call yourself an author and writer because it's true.