I believe the definitions are somewhat muddled, and have changed somewhat over time. For example, High Fantasy has been defined at various times as:
High fantasy is sometimes called epic fantasy, and some of the hallmarks of this subset of the fantasy genre include a high page count, lots of characters, usually a quest, and, most importantly, an alternative or secondary world as opposed to the real or primary world. With high fantasy, there are usual global stakes involved—you know, good versus evil, saving the world, and all that.
For a long time, The Lord of the Rings was held up as a standard bearer for High Fantasy, as it it the very ideal of the
Epic; what with a quest, global stakes, saving the world and the Good vs Evil. But most of us would probably refer to it as somewhat Low Magic. So I would agree that it is probably useful not to conflate High/Low Fantasy with High/Low Magic.
Yet Tolkien himself intended the story to be set in the mythical past of our world. He once stated in a letter that in Middle-earth, "miles after all are miles," indicating that Middle-earth followed the same general mundane rules of reality that we have today. Indeed, Bilbo and Frodo seem to live very mundane, prosaic real-world type lives (Hobbits & Hobbit holes notwithstanding), until adventure intrudes on them from Outside
. All of which sounds very Low Fantasy by the definition up thread. Of course, this is not precise, because Dragons, Wizards, Goblins & Elves are rumors and legends for the most part to the rustic Hobbits, until they show on Bilbo's doorstep.
So I would think it depends on what part of the definitions you want to cling most closely to: The Epic quest with Good/Evil battles & saving the world, vs how closely the fictional story world hues to our own. You could classify Middle-earth either way, depending on which characteristic you want to emphasize. I would probably characterize tLotR as High Fantasy (with a Low Magic bent), with Conan largely being Low Fantasy (much more amoral with treasure and personal gain involved, good vs evil is really very relative, not so much save the world as save yourself) despite fantastical elements in the world. I don't think I would classify Harry Potter as Low Fantasy, both because there are so many fantastical/magical locations & creatures and because it is much more of an Epic Quest, Good vs Evil, Chosen One vibe.