: Was it engraved on the Ring after the rhyme was composed or did someone compose the rhyme to incorporate the words from the Ring? And, if the latter was the case, how did anyone know what was written on the Ring?

I would suggest that the two lines relating to the One Ring were possibly part of the magick which bound the rings together. We know that Sauron was skilled in songs of power since he whooped Finrod during the first age. Maybe the enchanting of the One involved some sort of Song of Power to be incanted by Sauron and the two lines relating to the One were always meant to part of it.

The other lines could then have been added afterwards, after it was known to whom the other rings of power had been given. But these lines of the verse were merely a method of knowing to whom the rings went and they had no part in the enchantment that bound all the rings (lesser or otherwise) to the One Ring.

This seems plausable, it could well fit and does not require and inventiveness or addition to what we already know (or suppose).

Cheers

Padster