King of the Elves

[the phantom]

Didn't Finarfin himself hold the title (King of the Noldor) in the West (Valinor), once the host of Fëanor and Fingolfin were gone to ME? Or was he simply "ruler" of the Noldor who stayed behind and did not go to ME?

This is what the Silmarillion says-

...and Finarfin was set to rule the remnant of the Noldor in the Blessed Realm.

But the host of the Valar prepared for battle...and those also of the Noldor who never departed from Valinor, whose leader was Finarfin...

It seems he's not a king, but rather a leader and a ruler.

Gil Galad and his relatives before and after were only kings of the Noldor. They had no power over Thingol's people or the Green Elves of Ossiriland.

Gil didn't have power over them, but Elrond does. He descends directly from Thingol and Melian (Thingol->Luthien->Dior->Elwing->Elrond).

So it seems he could claim the kingship of both the Noldor and Teleri (he just never bothered too).