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I came across a jarring statement in Unfinished Tales, in the chapter "The Palantíri":

It must however be noted with regard to the narrative of The Lord of the Rings that over and above such deputed authority, even hereditary, any "Heir of Elendil" (that is, a recognized descendant occupying a throne or lordship in the Númenórean realms by virtue of this descent) had the right to use any of the palantíri.

This passage seems to occur "in-universe", which is why it surprised me. I don't recall coming across a reference to The Lord of the Rings as such in Tolkien's other works, in an in-universe context. I was also under the impression that Frodo's title for the story we know as The Lord of the Rings in The Red Book of Westmarch was considerably more wordy, something like The Downfall of the Lord of the Rings and the Return of the King.

Would Frodo, Bilbo, or Sam have called the story The Lord of the Rings? If not, did Tolkien simply make a mistake here?

The story is titled (in-universe) as:

             My Diary. My Unexpected Journey. There and Back Again. And
What Happened After.

                                Adventures of Five Hobbits. The Tale of the Great Ring, compiled by
                           Bilbo Baggins from his own observations and the accounts of his friends.

                                                      What we did in the War of the Ring.

                                                                      THE DOWNFALL

                                                                              OF THE

                                                                  LORD OF THE RINGS

                                                                            AND THE

                                                                  RETURN OF THE KING

                  _{(as seen by the Little People; being the memoirs of Bilbo and
Frodo of the Shire, supplemented by
                                                  the accounts of their friends
and the learning of the Wise.)}

                                       _{Together with extracts from Books of Lore translated by Bilbo
in Rivendell.}

It seems at least reasonably likely that they would shorten the name of the story to "The Lord of the Rings" since that's a pretty catchy title (as evidenced by the fact that at least one other author (Tolkien) calls it that).

Richard has correctly identified the in-universe title of the story. However, that has nothing to do with the reason that Lord of the Rings is referred to by that name in the "Palantíri" essay. Christopher Tolkien explains the source of this essay in the introduction to Unfinished Tales:

For the second edition of The Lord of the Rings (1966) my father made substantial emendations to a passage in The Two Towers, III 11 "The Palantír" (three-volume hardback edition p. 203), and some others in the same connection in The Return of the King, V 7 "The Pyre of Denethor" (edition cited p. 132), though these emendations were not incorporated in the text until the second impression of the revised edition (1967). This section of the book is derived from writings on the palantíri associated with this revision; I have done no more than assemble them into a continuous essay.

(emphasis added)

In other words, this is not a slip on Tolkien's part; he's writing this essay as the author, to try and clarify his thoughts on a section of his book.

The book itself, or at any rate the manuscript conceived of as being the basis of the novel, is referred to by name only once by any character. As Frodo is consoling Sam at the Grey Havens, he tells Sam:

You will be the Mayor, of course, as long as you want to be, and the most famous gardener in history; and you will read things out of the Red Book, and keep alive the memory of the age that is gone...

That's the only reference to the book, by name, by a character in the book.