"I wonder if the practice of the USA being called "America" originated in the USA or happened elsewhere and got taken up in the USA subsequently? Or was it the other way round?"

Good question, Kevin, and interesting food for thought.

I think that this convention has its origins in England. Way back in the time of Elizabeth I, when the first explorers and settlers were making their way to the New World, they would quite naturally be said to be "going to America." The fact that English citizens and other English-speaking people generally went to one particular large area of North America, while the Spanish and Portuguese gravitated towards points south, was taken for granted.

English folks had no need to say, or to think, that their friends and family (and/or their prisoners) were moving to a different part of America than other Europeans.

Subsequently, of course, the English/British/Celtic English-speaking colonists thought of themselves, and called themselves, "Americans." Again, they were living in a different, smaller world than the one we inhabit today, and there was always an unconsicous assumption that there was never any reason to differentiate "English America" from the very distant areas to the south, or for that matter even from the much closer environs of Francophile North America.

In summary, this usage may have originated at least partly "in the USA," geographically speaking, but long before the area in question became the "USA" ~ rather, it originated way back when the area was the "British Colonies." I would propose that it originated among the English-speaking British on both sides of the Atlantic.