But it's not. North America and South America are just as separate as Africa and Eurasia.

>sticking to definitions that are internationally agreed upon.

"American" is used as the demonym pretty much everywhere in the world other than Central America and South America. So that advice is backwards.

And try to empathize by imaging that you can't call yourself a demonym, and you have to either go by a bizarre acronym like "RAian" for the Republic of Argentina, or when someone says "I'm French, what about you?" you have to say "Well, I am living in the Republic of Argentina"

Then turn around and direct this at Australians. They apparently have no right to their demonym either, because New Guinea is part of the continent. And in their case, it's actually part of the continent.

https://en.wikipedia.org/wiki/Australia_(continent)

Papua New Guineans and Indonesian Papuans already have their own names for themselves, but what if they also want to say they're Australian? Let's have everyone switch from "Australian" to "CAian" for the Commonwealth of Australia.

San Marino is literally on the Italian Peninsula, so "italiano" is off completely. They need to start going by "RIano" or something.

And how many countries are in the South of Africa?

Solely by misappropriation. America was world-wide known as a continent before England sent their colonizers. Somehow migrating to America was misappropriated as meaning "migrating to the English colonies in America". Somehow the United States of America became the only America.

I have the theory that if their name had been less generic, say United States of New England, The word America would have retained its original meaning.

Could you clarify that? I understood that as both cases are separated by a channel (Panama vs Suez), or did you have another meaning?

[0]: https://en.wikipedia.org/wiki/Demonym

I'm aware that there are some continent categorizations that group North and South America as one continent called America, so one could use American to mean people who live in one of what are understood in other categorizations as North or South America. (Is that the continent categorization that you're familiar with? I just recently came across this, so if you wouldn't mind sharing where you learned this, and whether you're familiar with categorizations that count North America and South America separately, I'd be interested to hear more about it.)

That doesn't, however, mean that the usage of American to mean citizen of the US is wrong: it's very widely used in English worldwide. USian is assuredly not widely agreed upon as an alternative, at least in English which is the language of discourse here. Words have multiple meanings that need to be understood in context. Sometimes that context needs to be clarified.

Who came to the agreement that "American" is only a denonym for someone who lives in the continents of North and South America and doesn't refer to someone from the US?

Look at how people from the US are referred to in some international languages: un américain, Amerikaan, Amerikaner, amerika-jin, Amerikano, etc.

America is a continent, and US citizens are not more American than any Canadian, Argentinian or Peruvian is.

Someone telling a Canadian they aren't American is a completely different concept from referring to someone from the United States of America as an American. Quite frankly, your argument doesn't follow.