I like to consider myself a writer, but that's just the closest thing I can think of to call it. I'm not great at writing. I'm okay with grammar and stuff, but even if you have horrendous spelling or grammar, that shouldn't stop you from writing. You might find that your spelling and grammar gets better with time as you write more.
What I like to think I am is a story teller, and my preferred method of telling the story is by writing it out. My goal isn't even to write to share with others, but write it so I don't forget it. I have a friend who is an english major and after telling him a few things about my writing, he was impressed at my ability to flesh out and make things make sense between all the different aspects of a story(plot, setting, themes, characterization, effective use of hot topics etc...) When it comes to actually writing it out in a format that someone would read, I find that I'm not so good at that. What I write usually reads like a mix of a general 2nd person outline/description of events(think police report), with some first person, and on occasion some dialogue mixed in when I know very specific things and the occasional info dump or notes in parenthesis. It's not something that would ever be published in a book as it is, but it gets the story documented, and out of my brain. I struggle with trying to format my writing so it's in a common format like 3rd person omniscient, or 1st person, and usually give up because it's mentally draining and doesn't come easy. Regardless, I still write out the story in my own way.
Last night as an exercise, and I was all by myself when I did it, I verbally went over the story I've been working on as if I was describing it to someone else and I caught two problems that dealt with the logical flow of the story.
I avoid structured writing like the kind of stuff you do for schooling, research, and work like the plague.
Overall, you just write what you like, and if you don't have a problem writing what you like, then you're a writer. I don't have a problem writing down my stories for myself, as I write for myself, and I'm a writer.