And if Game of Thrones is low fantasy and yet it has similar amounts of magic as Lord of the Rings, then why don't we call Lord of the Rings low fantasy too?

For the same reason we don't call steaks and briskets the same thing. They're both beef, but what you do with them matters. The magic in Lord of the Rings tends to be wondrous and spectacular. It's entirely in keeping with the themes and motifs of high fantasy epic: a wizard riding into battle against the forces of the Dark One, staff held high to blind the Nazghul. Or a magic ring with the power to dominate all life, everywhere, forever. The story is an epic battle between good and evil and the magic reflects that. Even little moments of magic, like Gandalf intimidating Bilbo, serve to help reinforce the theme of a battle between good and evil for the fate of the common folk.

But Game of Thrones isn't really an epic. It's easy to mistake for one. There are dragons, and an invasion of icicle zombies, to worry about. But characters specifically and deliberately elect to ignore those magical threats. King Robert calls off Danaerys's assassination, and later on the Lannisters openly mock the Night's Watch's requests for more men to guard the Wall. Those decisions will come back to bite Westeros in the ass, but the decisions were made in service to the story. The story is about people, not Westeros.

The "holy shit" moments of magic impinging on the War of Five Kings plot helps drive that theme home. Melisandre and Caitlyn are the two biggest examples. In other cases, magic has little to no impact on the story, at least not so far. The warg connection is a great example of this: most of the dire wolves die or disappear without making a dent in the plot. Two of the counter-examples are Bran and Anya. However, Bran and Anya also wind up physically removed from Westeros, pushed to the war's sidelines. I honestly don't know how either of them are in a position to make any impact on the plot at all, as of the end of book 5.

Another example of the futility of trying to use magic to save the world is Anya's stint with Jaqen H'ghar. She wastes the opportunity, and the narration straight-up says so.

You're right that the difference between high and low fantasy isn't in how much magic there is. It's in the scope of the story: high and wide or low and tight.

I'd like to ask you a question. Way back in my first post, I said GoT took place in a high fantasy world. I've reiterated that in each of my follow-ups. So

Lord of the Rings is considered high fantasy, while Game of Thrones would never be considered that.

Wat?