living dead vs dead person

living dead and dead person both are nouns.

Word	Noun	Adjective	Verb	Adverb
living dead	Yes	No	No	No
dead person	Yes	No	No	No

As nouns, dead person is a hypernym of living dead; that is, dead person is a word with a broader meaning than living dead:

living dead: a dead body that has been brought back to life by a supernatural force
dead person: someone who is no longer alive

Other hypernyms of living dead include dead soul, deceased, deceased person, decedent, departed.

living dead (noun)	dead person (noun)
a dead body that has been brought back to life by a supernatural force	someone who is no longer alive