There is a tendency to think of music as either a rarifed occupation given to certain so-called talented people or as a commodity. This gives rise to questions such as these. Then there is the problem that some consider what they like as music and what they don't like as noise.For example, is Barry Manilow a musician? I think he is but that he wouldn't pass muster amoung many folkies I don't think. Then, is Dock Boggs a musician? I think so but so many of my fellow musical friends would take exception to that.
Then there is the whole thing about being a "trained" musician. Does knowing how to write five-part convertable counterpoint make one a musician? Probably.
I think that the key to being a musician is probably communication although here it gets sticky, too. What communicates to one person may not be to another.
One of the interesting aspects of people who I would consider to be musical or musicians is their ability to appreciate all forms of music to some extent. They have the ability to find the music in what others may characterize as noise. I think you can appreciate music without even liking it that much. For example, I admire Jimmy Page and Arnold Schoenberg for their musical ability but I don't really care for their music. Nor do I like Wagner all that much but I'm not going to say that he was not a great musician. As Twain said, his music is better than it sounds.
The love of music to me is characterized by a kind of open-ness and appreciation for all kinds of music. I think of Charlie Parker's statement, "Let's don't call it jazz, let's call it music."
Frank