Am I a writer?

I spend my spare time thinking about stories. I even sit and write in a notebook. There was that great story I wanted to tell, so I pulled out my notebook and grabbed my favorite pen.(something I prefer to rather than starring at a blinking cursor on a computer screen) I'm all set. I take the top off the pen, open the notebook, when to my surprise the last page has already been used.
I've heard many writers say "You're not a writer unless you write everyday".
I think to myself, "I must be a writer then, I write everyday". Wait, they must be wrong. The authors I've admired, aspire to be like. Do they know what they're talking about?
It takes so much more. An idea, patience, willingness to tell others a story, and commitment. There's so much more involved in telling a story.
Especially one to be proud of. To engage the reader, have them wanting to read that whole story.
I don't expect to be a Patterson, Grafton, or even a Grisham.
All I want to do is write something someone would find interesting.
Have you heard another saying (quote) "Better to write for yourself than write for the public and have no self" ? That's by Cyril Connolly