In my books, a musician is a music addict, usually multi-genre or at least open minded to all styles.Someone who would just as much pick up a tennis racquet as pick up a guitar is not a musician.
However, I would include in this special family of 'musicians' those people who are compulsive listeners/collectors but have never learnt to play an instrument.
It's a most wonderful thing when you mention a song or a piece of music which you love or which helps define you, and although you thought it was obscure, someone else in the room knows and loves it.
--Callie