major leaguer vs ballplayer

major leaguer and ballplayer both are nouns.

Word	Noun	Adjective	Verb	Adverb
major leaguer	Yes	No	No	No
ballplayer	Yes	No	No	No

As nouns, ballplayer is a hypernym of major leaguer; that is, ballplayer is a word with a broader meaning than major leaguer:

Other hypernyms of major leaguer include baseball player.

major leaguer (noun)	ballplayer (noun)
a member of a major-league baseball team	an athlete who plays baseball