(This is a bit of an overreaction, probably. Most of the time, people are not this snippy about it. But it might help some Yanks understand how annoying certain usages are to other people.)
It's "the United States of America" because those that named it meant to annex the whole of "America." It was the doctrine of Manifest Destiny. I imagine many gringos still hope to subvert Canada fully into the USA, even if they've largely given up on absorbing Latin America. (This explains much of the response to Ted Cruz's Canadian birth.)
The USA usage directly derives from this historical political stance. Defending it now because, "that's how we use it, so you latinos can suck it and accept it," is silly. All of these other people use it the other way, so gringos can suck it up and accept that. Do you mean to revive Manifest Destiny and the conquest of the hemisphere by a band of self-important Anglo-Saxon republicans?
Note that many Europeans still think that "America" is one continent, incorporating what we call "the Americas." And some are barely aware of the rest of the hemisphere and just assume that they are our unimportant satellites, rather like gringos may see Sri Lanka, Bangladesh, and Nepal as just parts of India. http://satwcomic.com/world-view
The usage of "American" to mean "gringo" is not just an insult, but an attempt to mentally erase the identities and sovereignties of all those who have struggled to be independent of gringo imperialism. It is a meme-crime.
The colombiana in the story didn't say that Mr Kirk couldn't call himself an American. She said that he couldn't redefine the word to stop her from calling herself American, because she is. Mr Kirk is defending the everyday erasure of América in favor of "America." Dead wrong, and racist.
So call it Yankea, Gringostan, or the territory of the blood cult of Georgie Washington and his liberated slavers, but if you call it "America" and then tell people from actual, real America that they're not Americans, getting scolded is least you deserve.