I had an argument with my wife once when I was editing a paper from Estonia which dated the Iron Age to the 10th c. AD; she said that was too late. Too late for Italy, perhaps, or Germany (if Germany ever actually had an Iron Age) or some parts of southern Africa, but... not wrong for Estonia.
Similarly there are parts of North America which aren't "historical" until around the middle of the 19th c.
Bowdoin van Riper also had a good definition of "historical archaeology" as something practiced by some post-antiquaries in England from the mid 1840s-mid 1860s: people who studied Roman, Saxon, medieval and post-medieval remains, and provide a contrasted with the nascent prehistorians.
There are also legal definitions (especially in Scandinavia) where monument protection is extended to everything "prehistoric," with "history" being defined according to a certain date. In Bavaria, for example, generally only "prehistoric" sites are protected while everything from Roman to concentration camps more or less has to be argued on a case-by-case basis. In Saxony they're also trying (or have perhaps succeeded) in removing protection for "historic" sites.
Maybe a more useful distinction would be prehistorical archaeology/historical archaeology, and "North American historical archaeology," seeing as the "historical" depth in the Americas & Australia is so much shorter than in much of the rest of the world. After that, it would seem that if someone is trying to postulate a "qualitative difference" between the world post ca. 15th century, then some sort of cause for such a qualitative change should be discernable. The implication seems to be that the discovery of the "New World" has something to do with it, Marshall McLuhan made an argument about how mass-production of moveable type opened the doors to the "Gutenberg Galaxy," a significant change in the means of production and ensuing ideology/world view, but we could also point to the aftermath of the Crusades and the Renaissance, or the Protestant Reformation, etc. After that we might start looking for some kind of Foucauldian "rupture," bearing in mind that Foucault based much of his framework on French examples that don't necessarily translate well into other European traditions.
Otherwise, without identifying some kind of cause, we're just looking at classification, semantics, reification... and that's just going to result in arbitrary divisions and/or labels.
I am still shocked by such a dogmatic response from people who were presumably trained in the American anthropological tradition (4 schools) of archaeology. I posted the link because it seemed a useful little video which could be shown to prospective clients or funding agencies who might otherwise balk at the idea of having to fund excavation & screening of latrines, cesspits, etc., particularly because Pompeii is so important to the history of archaeology (the site itself has now become an example of "historical archaeology") and is generally familiar with the "lay folk" who might be responsible for making funding decisions.

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Telling these colleagues that they should use the term (rather than the concept of) 'historical archaeology' in a specific sense just because North Americans and Australians do is not likely to win many converts to that usage.Summed up, I think it might be worth making a distinction between the recognised existence of an archaeology of the modern world - which surely few contributing to HISTARCH would dispute, even if they might disagree about the precise date boundaries thereof - and what the best name for that archaeology of the modern world might be.   They are wholly separate issues.