Your second point is just as good. For whatever reason, America claimed for itself the name adopted by Amerigo Vespucci, "America." I mean, I'm half Ecuadorian myself and even a very Castillian surname, but it doesn't bother me. Then again, I'm an American, so I don't know how South or other North Amricans feel about it.
I've lived extensively in both Central and South America. There appears to be a slight difference in outlook between the two. In Central America, they refer to themselves in the collective as either "centroamericanos" or "americanos". In South America, they refer to themselves as "latinoamericanos" - a grouping which they insist doesn't include other latin americans such as the Central Americans. I guess everybody has their issues on this particular subject.
As far as I can tell, Native Americans would like to be referred to as their tribe affiliation. A Hopi does not consider a Navajo to be the same, even though they likely have similar DNA. But I guess its the same as why someone from Norway wouldn't want to be affiliated with Swedes, even if they share Nordic traits.
You are 100% correct. Interesting that you should mention the Hopi and Navajo. I'm originally from the southwest, and the bare tolerance between the two groups is noticeable. They have very different cultures, and don't even speak a language from the same language family (Navajo is Athabaskan, Hopi is an Uto-Azteca derivative). I learned early on not to make the mistake of lumping all indigenous groups into one collective pot. The Dineh (literally, "The People") would be extraordinarily insulted to be mistaken for the Hopituh (literally, "The Peaceful Ones"), and vice versa. In this context, "Indian" isn't any better than "Native American", IMO.