When I was a kid Roger Sprung once snarled at me to, "think like a musician instead of a banjo player", and years after that my music theory teacher in college said that I was, "a great banjo player but a lousy musician."
Both comments confused the crap out of my back then.
I've been making music for for than twenty years now. I make a decent living teaching banjo & guitar and from time to time I find myself snarling at somebody to, "think like a musician instead of a banjo player."
What does that line mean? Well, the answer really depends on the student. I think that the line between somebody who plays a few songs on the guitar or the banjo and a person you could think of as a "musician" is really a matter of perspective. To a person who can't play anything a guy who knows three tunes on the banjo looks like a genius.
In the end I look at the term musician as a way to describe a person who is comfortable enough with his or her chosen medium to work in a variety of settings without a lot of fuss and bother - but since the term is all based on perception I'll probably think it means something completely different next week or next year.
It's just a word. How people describe you and define your music shouldn't mean that much to you. You is who you is, and if you've got that part under control then "who you is" is going to change and grow as you mosey along.
Just make music and be yourself. Judging yourself against other people is always going to leave you feeling inadequate in some way or another. I'm a good banjo player and I love what I do, but if I compared my achievements and training to that of a concert violinist I could start to feel inadequate. What we forget is that the violinist in question may also be comparing himself or herself to somebody else. Don't be distracted from your own personal journey by falling into this trap. Allow the learning process to work.
And keep in mind that the learning process never really ends. We are al, each and every one of us, a work in progress.