red man vs Amerindian

red man and Amerindian both are nouns.

Word	Noun	Adjective	Verb	Adverb
red man	Yes	No	No	No
Amerindian	Yes	No	No	No

As nouns, Amerindian is a hypernym of red man; that is, Amerindian is a word with a broader meaning than red man:

red man: (slang) offensive term for Native Americans
Amerindian: any member of the peoples living in North or South America before the Europeans arrived

Other hypernyms of red man include Native American.

red man (noun)	Amerindian (noun)
(slang) offensive term for Native Americans	any member of the peoples living in North or South America before the Europeans arrived