I would probably characterize tLotR as High Fantasy (with a Low Magic bent)
A Low/Narrow magic setting, where few people have access and little, if any of it, is powerful, is what I consider Middle-Earth to be.
Magic in The Lord of the Rings/Middle-earth can be very powerful. Sauron causes whole armies to experience fear or battle rage at a distance of about 200 miles. He controls the weather 900 miles away. Galadriel, using Nenya, one of the Rings of Power, can control the weather and manipulate time over all of Lothlórien. Saruman impedes Aragorn and his companions while speeding up the orcs bearing Merry and Pippin at a distance of up to 300 miles. The Ent-like Huorns are sufficiently numerous and quick to create a forest where there was none before. They can also cloak themselves in shadow. Aragorn commands an army of ghosts. The dust Galadriel gives to Sam causes many changes to occur when it is planted in the Shire which,
according to wikipedia, is 120 by 150 miles across.
The following quotations are all from The Lord of the Rings.
Sauron mentally influences armies
These are the words of Boromir:
Sudden war came upon us out of Mordor, and we were swept away. We were outnumbered… but it was not by numbers that we were defeated. A power was there that we have not felt before.
‘Some said that it could be seen, like a great black horseman, a dark shadow under the moon. Wherever he came a madness filled our foes, but fear fell on our boldest, so that horse and man gave way and fled.
Sauron’s weather control
‘I wonder if this is a contrivance of the Enemy,’ said Boromir. ‘They say in my land that he can govern the storms in the Mountains of Shadow that stand upon the borders of Mordor. He has strange powers and many allies.’
‘His arm has grown long indeed,’ said Gimli, ‘if he can draw snow down from the North to trouble us here three hundred leagues away.’
‘His arm has grown long,’ said Gandalf.
"That day the weather changed again, almost as if it was at the command of some power that had no longer any use for snow, since they had retreated from the pass, a power that wished now to have a clear light in which things that moved in the wild could be seen from far away."
Galadriel's power over Lothlórien
"All the while that they dwelt there [Lothlórien] the sun shone clear, save for a gentle rain that fell at times, and passed away leaving all things fresh and clean. The air was cool and soft, as if it were early spring, yet they felt about them the deep and thoughtful quiet of winter."
Frodo: "[W]e were in a time that has elsewhere long gone by. It was not, I think, until Silverlode bore us back to Anduin that we returned to the time that flows through mortal lands to the Great Sea… The power of the Lady is on it. Rich are the hours, though short they seem, in Caras Galadhon"
Saruman slows his enemies and speeds his minions
I am weary as I have seldom been before, weary as no Ranger should be with a clear trail to follow. There is some will that lends speed to our foes and sets an unseen barrier before us: a weariness that is in the heart more than in the limb.’
‘Truly!’ said Legolas. ‘That I have known since first we came down from the Emyn Muil. For the will is not behind us but before us.’ He pointed away over the land of Rohan into the darkling West under the sickle moon.
‘Saruman!’ muttered Aragorn.
The Huorns
"The land had changed. Where before the green dale had lain, its grassy slopes lapping the ever-mounting hills, there now a forest loomed."
These are the words of Pippin:
‘There is a great power in them, and they seem able to wrap themselves in shadow: it is difficult to see them moving. But they do. They can move very quickly, if they are angry. You stand still looking at the weather, maybe, or listening to the rustling of the wind, and then suddenly you find that you are in the middle of a wood with great groping trees all around you.
Aragorn commands an army of ghosts
Legolas: "But Aragorn halted and cried with a great voice: “Now come! By the Black Stone I call you!” And suddenly the Shadow Host that had hung back at the last came up like a grey tide, sweeping all away before it."
Galadriel's dust
Sam planted saplings in all the places where specially beautiful or beloved trees had been destroyed, and he put a grain of the precious dust in the soil at the root of each…
His trees began to sprout and grow, as if time was in a hurry and wished to make one year do for twenty...
Altogether 1420 in the Shire was a marvellous year. Not only was there wonderful sunshine and delicious rain, in due times and perfect measure, but there seemed something more: an air of richness and growth, and a gleam of a beauty beyond that of mortal summers that flicker and pass upon this Middle-earth. All the children born or begotten in that year, and there were many, were fair to see and strong, and most of them had a rich golden hair that had before been rare among hobbits. The fruit was so plentiful that young hobbits very nearly bathed in strawberries and cream... And no one was ill, and everyone was pleased, except those who had to mow the grass.
In the Southfarthing the vines were laden, and the yield of ‘leaf’ was astonishing; and everywhere there was so much corn that at Harvest every barn was stuffed. The Northfarthing barley was so fine that the beer of 1420 malt was long remembered and became a byword.