[It's important to] know the names by which each Native tribe refers to itself and that you use said nomenclature when speaking of them rather than the names given to them by their enemies?
For instance, all of my "Navajo" students called themselves Dine' ("the people"). A few others are:
Apache ("the enemy") = Ndee ("the people")
Comanche ("they fight with us") = Numinu ("the people")
Mohawk ("man-eaters") = Kanienkehaka ("people of the flint")
This is my experience as well. If you go into the governmental offices of one of the Tewa pueblos of the Rio Grande Valley area, you'll see a seal with their name for their people in their language.
Thanks for posting Michael.