ball carrier vs football player

ball carrier and football player both are nouns.

Word	Noun	Adjective	Verb	Adverb
ball carrier	Yes	No	No	No
football player	Yes	No	No	No

As nouns, football player is a hypernym of ball carrier; that is, football player is a word with a broader meaning than ball carrier:

ball carrier: (football) the player who is carrying (and trying to advance) the ball on an offensive play
football player: an athlete who plays American football

Other hypernyms of ball carrier include footballer.

ball carrier (noun)	football player (noun)
(football) the player who is carrying (and trying to advance) the ball on an offensive play	an athlete who plays American football