A music director implies the person who selected pieces of music that other people wrote. Either selected the music for the band to play (in which case she probably also a bandleader.) Or she selected recordings for the film soundtrack (in which case the person could also be considered a producer in the sense it used in the music world -- though not in sense "producer" is used in the film world.)
A composer writes a musical score, which as noted above, usually does not include lyrics.
Scorer is less common, because as noted "score" now implies several slang meanings which have nothing to do with music :) (There is an album of film music with the humorous title "I Like to Score.")
Musician, as noted, means someone who performed but did not necessarily write music. "Music composer" is redundant in any context.