To first begin by picking a nit, I believe the word you want is "archetypal".
I think "high magic" and "low magic" are terms that are often
used to describe campaign settings, but which are too vaguely-defined in common parlance to be
useful. So I, personally, would scrap them.
The other problem is that your list contains items which aren't mutually exclusive - for example, it's possible to add "evil has already won" to any of the other campaign types you've mentioned. What we need to do is decide what we're talking about - are we discussing genres (such as "high fantasy"), tones (like "grim-and-gritty"), ground rules (for instance, "the nature of magic"), or a combination of all three?
I would start from the bottom.
There's a kind of "fantasy" that I would call "ahistorical" - that is, it's lacking in fantasy elements like monsters and magic, but attempts to reproduce (to varying degrees of authenticity) the historical world without being actually
set on Earth. You have castles and knights, peasants and kings (or helots and tyrants, samurai and daimyo, tribesmen and chiefs, whatever), but they don't live in England, or Greece, or Japan, or any historical Earth locale. History is fictional but produces "historical" Earth structures, or relatively close analogues.
A close relative is "alternate history" or "allohistory", which takes place in a world like historical Earth until a certain point, whereupon history diverges as the result of a change in events - for example, if the Crusaders somehow never lost Jerusalem and established a European-dominated Christian state in the Middle East which endured the centuries, or a campaign which takes place in the far-off border domains of Alexander the Great as the king approaches his eighth decade.
Then you have "magical allohistory", which is the above where the introduction or re-emergence of magic is the point of divergence.
"Historical fantasy" usually isn't very fantastic in the traditional sense, but many gamers get satisfaction out of accurate historical settings and adventure within them - outside of the adventure-oriented heroism of most D&D games, you'd find historical games leaning towards politics quite often, which of course could easily lead to an allohistorical setting if the players really manage to change things, and if the DM uses genuine historical circumstances as her starting point.
There's "mythic fantasy", which takes place in a world defined and shaped by the myths of one or more ancient cultures such as Greece or Egypt or China - and, arguably, a medieval game which took the beliefs of European Christians of the time about the structure of the world would fit into this category as well.
There's "epic fantasy", wherein the heroes (or at least the protagonists) end up battling for the fate of the world - either to save it, control it, or destroy it in one way or another.
(These terms are themselves amorphous - a game of mythic Norse fantasy could also easily be epic fantasy if mortal heroes must range themselves beside (or against) the gods at Ragnarok. As noted above, a historical campaign could easily become allohistorical if the players' actions have a significant-enough impact on the setting. It's also trivial to invent an ahistorical setting based on historical myths, such as
Nyambe.)
While D&D describes itself as "heroic fantasy", that's not really a genre
per se in my mind - rather, it's a phrase describing tone and expectations. Characters will be competent enough to succeed, but they will overcome challenges worthy of heroes (or at least people of heroic stature, though for marketing reasons I suspect Wizards uses "heroic" for its connotations of right moral action as well).
"High" and "low" fantasy, like "high" and "low" magic, can be used in different ways depending upon the biases involved, and aren't necessarily synonyms for each other. A game of mythical Greek adventure, for example, could be high fantasy, chock full of weird monsters drawn from those myths - but it could be low magic, if the only magic available comes from the gods, prophets, and hermits of the world, and the only magical items are the property of these beings loaned out to worthy heroes. As I said before and elsewhere, I don't think they're useful descriptors - it's much better to be explicit about what goes on in a given setting than to use obfuscating shorthand.
Some people might choose - though I wouldn't, at least not quite as readily - to contrast "heroic fantasy" with "grim and gritty fantasy"; if the terms weren't so vague you could also use "high" and "low" fantasy for this purpose, here turning on the extent to which success has what you might call a mundane price. Grim-and-gritty fights are usually much deadlier, even when they're relatively meaningless to the plot, compared to the heroic cutting-down of mooks as your heroes head for the final showdown with the villains. Other examples prevail, and of course the terms are often so confused that people think any game with weaker magic than the D&D norm is also "grim and gritty" . . . all these terms are useless!
