doorbell vs push button

doorbell and push button both are nouns.

Word	Noun	Adjective	Verb	Adverb
doorbell	Yes	No	No	No
push button	Yes	No	No	No

As nouns, push button is a hypernym of doorbell; that is, push button is a word with a broader meaning than doorbell:

doorbell: a push button at an outer door that gives a ringing or buzzing signal when pushed
push button: an electrical switch operated by pressing

Other hypernyms of doorbell include button, push.

doorbell (noun)	push button (noun)
a push button at an outer door that gives a ringing or buzzing signal when pushed	an electrical switch operated by pressing