Alternate names for unconquered Nations in America (+ Alternate colonies names)

So every once in a while we discuss about what would happen if the main indigenous nations of America managed to fend off the European settlers and conquerors, from the least (Aztecs) to the most (Incas/Tahuantinsuyus) plausible.

1.Something that I keep wondering, more specifically in Meso-America, about is how would these nations call their own countries. Because we get a lot of adjectives (Aztec Empire, Mayan Kingdom), but few actual names for the kingdoms and empires themselves. I know that the Aztecs use the term Aztlan to describe their place of origin, but it doesn't designate the Mexico valley.

So what terms would they likely use to call their own countries?

2.On the other hand, I know the incas used the term Tahuantinsuyu for their empire, but what gentile/adjective would they use to call themselves?

3. I also wonder what names would the European settlers take for their colonies if they do not manage to conquer these lands. The elephant in the room there being Mexico, being named specifically after the Mechica in Tenochtitlan. If Spain managed to settle the parts of the land in Latin America not covered by the Aztecs and Tlaxcalans etc... Would they just stick to New Spain until it declares independence? And when they become independent, would the take the name of a local people (e.g. Raramuro...)?

4. There's already been ample discussion about the alternate names of the "USA" and the "Americans", the most plausible alternative so far being the Columbians and something like United State of Columbia (not too sure about the latter), what other possible names would be used?
 
The Aztecs called their polity the Triple Alliance, or Ēxcān Tlahtōlōyān

The Incan Empire was indigenously characterized as the Four Regions, or Tawantinsuyu
 
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