Red Indian vs Amerindian

Red Indian and Amerindian both are nouns.

Word	Noun	Adjective	Verb	Adverb
Red Indian	Yes	No	No	No
Amerindian	Yes	No	No	No

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

Red Indian: a member of the race of people living in America when Europeans arrived
Amerindian: any member of the peoples living in North or South America before the Europeans arrived

Other hypernyms of Red Indian include Native American.

Red Indian (noun)	Amerindian (noun)
a member of the race of people living in America when Europeans arrived	any member of the peoples living in North or South America before the Europeans arrived