Wilfried, Thanks for the reminder about the adverb/adjective distinction. (My Latin classes were decades ago.)
However, "Adeste Fideles" was not written by "the old Romans." I believe it was composed in the 12th C. (or maybe later). I don't know if all publications of the song used "laeti," (instead of "laete") but the English version used to be "joyfully triumphant." If the Latin says "laeti triumphantes," wouldn't that translate as "joyful triumphant," not "joyful and triumphant?" I don't know if Latin speakers would say it that way, without the "et," but in English wouldn't you say "joyfully triumphant" rather than "joyful triumphant?"
At any rate, I would consider the lyric drift more an example of the folk process than a misquote.

My favorite example of lyric drift is the saying/song "Let us smile 'be[neath] your umbrella," which evolved into "Let a smile be your umbrella." In part, it was probably originally a mondegreen, but by now, no one [in the US, anyway] uses "be'" as a contraction for "beneath," so the original line would probably not be understood by most folks. Still, the two lines have quite different meanings.

Genie