designated hitter vs batsman

designated hitter and batsman both are nouns.

Word	Noun	Adjective	Verb	Adverb
designated hitter	Yes	No	No	No
batsman	Yes	No	No	No

As nouns, batsman is a hypernym of designated hitter; that is, batsman is a word with a broader meaning than designated hitter:

designated hitter: a ballplayer who is designated to bat in place of the pitcher
batsman: (baseball) a ballplayer who is batting

Other hypernyms of designated hitter include batter, hitter, slugger.

designated hitter (noun)	batsman (noun)
a ballplayer who is designated to bat in place of the pitcher	(baseball) a ballplayer who is batting