Actually the phrase is not "American natives" but "Native Americans".
I think what the writer of that definition was trying to say was that the word "native" as a stand-alone noun to mean a person from a non-Western culture with a low level of technology is now considered offensive. Like if you drew a picture of a group of people standing in front of a mud hut, with painted faces and carrying spears, and labeled it "Natives", this would be considered offensive.
The word "native" in general simply means some one or thing that originally comes from a particular place. In this sense it is a perfectly good word. I certainly would not be offended if you referred to me as "a native of New York". We routinely talk about "foods native to the region", about a person's "native language", etc.
But anyway, I don't think there's any simple rule as to what makes a word or phrase offensive. When I was a boy in the 1960s, members of a certain ethnic group were routinely referred to as "negroes". Then about the 1970s or so we were told that this term was offensive, and that we should call them "black". Then in the 1990s we were told that "black" is offensive, and we should call them "African-American". How did "black" go from being the polite term to being offensive? It just did. There's no pattern to such things.
I saw a survey a few years back that found that a majority of American Indians prefer to be called Indians rather than Native Americans. For that matter, I saw a survey fairly recently where they asked black people what they prefer to be called, and 1% said "African Americans", 2% said "black", 96% said "don't care", and there was the usual scattering of other answers.
May I take down your name is for people registering visitors on a piece of paper
May I have your name can be used in a formal manner
If you ask someone their name in an non-official name-taking place, you would use
What is your name?
May I get your name does not sound idiomatic to me unless you are planning to marry them and take their name for yourself
If someone introduces themselves and you did not quite understand it, you can say
I did not quite get your name, please repeat?
Best Answer
The Brits call themselves Brits.
It's just an abbreviation, not a derogatory term.
You could avoid it if being formal, but in everyday conversation it's fine.
You could further sub-divide into English, Scottish [or Scots... never scotch, that's a drink not a person] & Welsh, but you'd need to be certain which they actually were.
It's easier not to guess & just refer to them as British.