baseball field vs playing area

baseball field and playing area both are nouns.

Word	Noun	Adjective	Verb	Adverb
baseball field	Yes	No	No	No
playing area	Yes	No	No	No

As nouns, playing area is a hypernym of baseball field; that is, playing area is a word with a broader meaning than baseball field:

Other hypernyms of baseball field include athletic field, field, playing field.

baseball field (noun)	playing area (noun)
the baseball playing field	a piece of land prepared for playing a game