In Britain I used to call all natives of North America "Red Indians" in the 1950s.
About 20 or 30 years ago I noted that the modern acceptable term was Native Americans. Less than 10 years ago I came across the term "First Nations" - certainly not a term seen in British schools in the 1950s or 60s. I have no notion of the sporting prowess of those people, except to note (as told by my mother - she had a sister who had emigrated to Canada) that lacrosse is a Canadian game, possibly from native sources.
Once you get south of the USA they are simply "Mexicans" as a generality, then "Hondurans", "Costa Ricans" etc as appropriate. In the Caribbean they are still "West Indians" in my mind if they are not specifically Jamaicans, Haitians, etc.
In South America they are Peruvians, Brazilians, etc, or possibly lumped together as South Americans.