"Battle of the Peak"???

[Son of Númenor]

There was no official name for the battle, and if there were, it would have been created after Gandalf said the above quote, probably after the War of the Ring, for the lore-books.

[Boromir88]

Tolkien also rights in a lot of point of view, just because someone says something doesn't mean that person is right or wrong. Gandalf just calling it the "battle of the peak" might have just been him, doesn't mean everyone named it "the battle of the peak." But, it is very possible it was written in lore books and called "the battle of the peak" either way thats what it's referred to as I assume.

[dancing spawn of ungoliant]

At least Frodo might have used the name "the Battle of the Peak" in his book when referring Gandalf fighting the balrog. But this is what the Encyclopedia of Arda says about The Battle of the Peak:

(It is) Gandalf's name for his climactic battle with the Balrog known as Durin's Bane, on the peak of the mountain called Celebdil, the Silvertine.

Note: Gandalf's name. So they seem to think that it might have other names as well.