I actually wanted to start a whole thread called "High/Low Fantasy" just because people tend to throw the terms around and often meaning the exact opposite of what they say. Part of that has to do with the fact that it is very blurry line between high and low fantasy. As an example.
Conan is almost always put in the low fantasy side of the argument. The big idea here is that the setting is a prehistoric, but very much our planet Earth as a setting. Magic is rare, and most stories are on a simple scale, ie robbing a temple of an artifact, rather than stopping an Evil Wizard from taking over the world.
The opposite of this is LotR, which is typically seen as "high fantasy." You have many different species (humans, elves, dwarves, hobbits) existing in a complex political environment, ie "the elves and dwarves have been at war for thousands of years." Magic is commonplace, if you aren't a wizard you probably know one. The stories tend to deal with world changing events, ie, Good vs Evil.
Now, here's where it gets complicated. Although supposedly "rare," magic of some type occurs in most Conan stories. And although most of the "evil wizards" are local baddies, there are certainly a few who try for something like world domination. What I like about those cases though, is they usually conspire with human allies (kings, princes, etc) and use their powers to expediate what would otherwise be normal military conquests.
On the other hand, Tolkien insisted that "Middle Earth" really was our earth, just in the distant path, even with all those mythical races.
So you can see how the two genre's tend to get mixed and matched a lot. It's really hard to define. Generally though, if you have any society of mythical creatures living alongside humans, high fantasy. An exception can be made for dying races which only a few people have witnessed. But when elves, dwarves, and humans get together to form a political sub-committee, that's high fantasy.
Im sorry, but the thing is Im an avid fantasy reader, and theres no such thing as low fantasy. The "low" in the low fantasy thing you are describing is not the "opposite" of high fantasy. High fantasy is so termed because it is sort of arthurean, based on an epic, out of this world sort of quest, and usually involves a coming of age story set in a medieval setting (meaning by medieval a chivalrous kind of place, with feudal structures).
The reason Conan , which is sword and sorcery, doesnt save the world, or that there no over the top, fireball spewing kind of magic, is more a device Robert Howard used than a genre thing. Howard always wants to keep you guessing if magic is real or not (although many times it is of course very very real, and very creepy), or if the gods exist or not. He wants to blur the line between wants confortable, and whats simply inhuman and completely savage in its nature (in a divine sort of way). He did that a lot in Solomon Kane: magic was always this thing that happened on the borders of civilization.
As for not battling the Evil Wizard (which he did in any case, his nemesis being Toth Amon) to save the world, well, this is where sword and sorcery is a thing of its own. Its a pulp genre in its heart, so with that comes a certain direction. In sword and sorcery, its usually the story of a lone anti-hero, and its more of a personal adventure than a chronicle of a setting (like in Middle Earth).
Now, I disagree with your view of LOTR. Wizards were not commonplace at all, and magic was a sort of artistic thing. Before the War of the Ring, mankind thought hobbits were fairy tales, and most have never seen elves and in some places people might not even think they exist. So LOTR, paradoxically, is different in this respect to the D&D craze it originated.
EDIT: I actually looked it up in wikipedia...and the concept is simply laughable. Its a made up genre that some crackpot book journalist came up with. The things that are considered "low fantasy" its simply fantasy! This is a crazy term that has no content, and that wikipedia article places under low fantasy things that are already classified, even by the authors themselves. So this is a completely unnecessary category that completely confuses an authors personal literary devices with an actual subgenre. Insane.