[Mithalwen]

This is being a bit picky but it is "Three Rings for the Elven-kings under the sky". You have inferred this to mean that there are three kings but that isn't necessarily true.

Partly I think there is a little poetic licence on the part of the translator of the poem to allow for some "elegant variation" in vocabulary. Kings is used because there are also dwarf lords and the Dark Lord but I think elven lords would be more accurate for the time referred to.

The rings of power were created by the Noldor with Sauron and I don't think that the Silvan realms even with Sindarin Kings, are referred to.

Gil galad was the High King of the Noldor it was a bit of an empty title even in the second age. In Beleriand where the grandsons of Finwe had realms it was more appropriate but in the second age there was only Lindon then Ost in Edhil and then Rivendell. I think Celebrimbor was called Lord of Eregion rather than King and Elrond was merely Master of a Household. So as far as the ring rhyme is concerned the "kings" were Cirdan, Gilgalad (then Elrond), and Galadriel.