Ah, if the question is “why did the scribes of Gondor leave out Gimli?” then we’re looking at a whole different question.
Gondor was elvo-centric. Their native tongue was a slight corruption of Elvish (I assume Sindarin). Their ruler, Elessar, had not only grown up in Rivendell, but married an Elf maiden, and I’m sure Arwen became all the rage among the fashionably people of Gondor. Yeah, sure, Gimli led a bunch of Elves to Gondor to aid in the rebuilding, but to the people of Gondor the Elves would have been the important ones. They’d have no particular animus against the Dwarves, but also no particular fascination with them, and probably insufficient knowledge of them to understand their rich heritage of song (albeit in a tongue-twisting language that sounded like rocks cracking against each other). The scribes had a good poetic reason for excluding a verse from the East Wind, so why bother inventing one? Give Aragorn two, because, after all, he’s the King. Give the Elf one, because he’s an Elf, and they’re all cool and shit. But why bother with the Dwarf?
Well, I think Aragorn (and Gandalf) would have stuck with the original cover story which Frodo recorded – that is, that Boromir followed Frodo so he could try to take the Ring and attempt to save Gondor with it. They knew that in actuality Boromir had followed Frodo into the woods to try to persuade him to “blow the horn of Gondor” <wink, wink, nudge, nudge>. The Ring story is a much more dignified account of a fine warrior’s final hours and that other business is best overlooked.
The high language (spoken by royalty, and used for things like official proclamations) was a dialect of Sindarin, but the native tongue, what most Gondorians would grow up speaking, was Adunaic (sometimes translated as Westron), a sort of mix of Sindarin and Rhohirric (not unlike how English is a mixture of French and Anglo-Saxon).
Are you sure? When Frodo & Sam first encounter Fararmir and his rangers in Ithilien, and are left behind with a couple of gaolers/escorts/bodyguards while the mass of the group goes off to ambush some enemy forces, it is specifically noted that the Rangers go from speaking Westron to a language Frodo recognizes as Elvish. (Which I took to be Sindarin.)
Yeah, but those are rangers, of higher blood to begin with, and they’ve probably had a lot more contact with the Elves than most Gondorians, too. And they were speaking Elvish specifically in the expectation that their captives wouldn’t know it. I think that there’s a bit, though, when Pippin is entering the city, that’s it’s remarked that the Old Language isn’t spoken much any more.