What a beautiful letter from Kimberley O'Brien!  Makes you love those Irish.
Jim, hard to say whether America's indigenous people actually  prefer "Native American" or "Indian",   Most probably prefer  going by their tribal name, as Kalapuya, Kalawatset, Siuslaw, Molalla, Umpqua, or even their band name as Chafin or Chela Mela. 
 In any case basic respect is that we call folks by the name they prefer and we don't use it in a disrespectful matter.  That's not PC,  just common decency..
Roseburg  HS, which has worked harmoniously with the local Cow Creek Indians to insure respectful behavior has kept the name Indians, while Marcola, whose "Indian"                                                                       name came from the local Mohawk -- an eastern term of no relevance locally-- chose to drop Indian. 
Fortunately we no longer picture America's First People as ignorant savages from whom we bravely "Won the West" as we did back when I was young, though that image lingers on.