Tolkien is almost low-magic. His work is also not quite in the "dark fantasy" setting to boot. Surely for the time, his writing could be considered dark fantasy, as prior to Tolkien elves and goblins and dwarves were all nearly benign sprites that were a part of childrens' folklore. The Hobbit was childrens' fantasy, and LOTR was more or less a coming of age tale in a way.

The Silmarillion was true high fantasy, borrowing elements from classical greek and late christian mythology. By the time the third age rolled around in Tolkien's world of Arda, much of the influence of the gods had faded from Middle Earth.

But here's what disqualifies LOTR from being low fantasy:

• Frodo and co. are walking around with a demigod (Gandalf)
• The main antagonist is a minor god (Sauron)
• The secondary antagonist is a demigod (Saruman)
• Elvish magic is still quite strong. (See the prayer of Elbereth, the ring-bearers, the enchantments surrounding the land of Rivendell)
• The line of Numenor still posess many godly boons.

Dark fantasy emerged from Tolkien's work. Tolkien's work created the bedrock of modern fantasy, and Gary Gygax's creation of Dungeons and Dragons made modern fantasy mainstream in the 1970s. It's no coincidence that almost all of these movies I listed were from the late 1970s to the 1980s. Because this was the peak of D&D's popularity, and at the time, these sorts of movies were ideal to produce owing to a lack of CGI technology at the time, and the high cost of conventional special effects. By eliminating fantastical creatures and pushing the influence of magic from fireballs and magic barriers to invisible effects like mind control, hexes, and curses, the Dark Fantasy setting was able to create a rich atmosphere with a low budget.

The modern LOTR/Hobbit films only come close to Dark Fantasy because of their production and direction, not because of their source material. Peter Jackson's influences include the Dark Fantasy genre because his formative years were during the height of Dark Fantasy's popularity.

In fact, Dark Fantasy had so thoroughly replaced high fantasy that Harry Potter (A true high fantasy series) managed to completely reinvigorate the genre that first inspired mainstream fair.

So yeah, the reason I don't count Tolkien's work, and by extension Jackson's work in dark fantasy has to do with the reasons dark fantasy emerged in the first place. Tolkien's work cannot be considered dark fantasy because dark fantasy emerged as a simultaneous contrast and reference to Tolkien's style.