personal check vs bank check

personal check and bank check both are nouns.

Word	Noun	Adjective	Verb	Adverb
personal check	Yes	No	No	No
bank check	Yes	No	No	No

As nouns, bank check is a hypernym of personal check; that is, bank check is a word with a broader meaning than personal check:

personal check: a check drawn against funds deposited in your personal checking account
bank check: a written order directing a bank to pay money

Other hypernyms of personal check include check, cheque.

personal check (noun)	bank check (noun)
a check drawn against funds deposited in your personal checking account	a written order directing a bank to pay money