The classic story is that of Grey Owl" which is told in that link.
I tend to be a bit suspicious of using names as a tool for changing social attitude - I mean leave aside ones that set out to be insulting, which some do. Indian doesn't set out to insult - it just reminds people of an embarrassing mistake made by European exploiters once upon a time.
"Red" - now that really is a bit strange, because the only people I've ever seen who actually have been bright red have been "white" North Europeans or of North European ancestry . But again, it's hardly an insulting term in itself. The insult doesn't lie in the words, it lies in the attitudes and assumptions of some of the people who use the words; and these aren't altered just by switching to a new label.
If I was choosing a new label in this case, I think the one I'd go from might be "Real Americans", in the hope of maybe pissing off a lot of the people I referred to in that last sentence. Except that it'd also probably upset a lot of good people as well.