It's interesting how people outside of a group can erase that group's identity just by taking away the name that they use to define themselves. Any politically correct American has to say "Native American" or use the even more generic term "Indigenous People" while the majority of the people being referred to understand themselves to be "Indians" or "American Indians". Hundreds of years being known as Indians and having that taken away by scholars and academics. It's a sad final twist on an exceptionally sad story.
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What is the correct terminology: American Indian, Indian, Native American, or Native?
All of these terms are acceptable. The consensus, however, is that whenever possible, Native people prefer to be called by their specific tribal name. In the United States, Native American has been widely used but is falling out of favor with some groups, and the terms American Indian or indigenous American are preferred by many Native people.
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This also provides a good overview through a series of personal viewpoints:
https://web.archive.org/web/20170913022941/https://indiancou...
Please point to these rights and legal theories. Do you just mean NAGPRA?
As far as I know, whatever legal rights any native American people have in the Americas at this point in time are based purely on them being here when Europeans arrived near the start of the 16th century.
Whether they had been the occupants for 400 years or 40,000 years wouldn't make any difference to the treaties that were signed (and generally abrogated).