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I don't see the point; people will have questions about who the Gamayuns are, they'll Google it, and it's a cringe-worthy, wingless, legless bird) + it doesn't sound as intimidating.
And Gveykhira also sounds cringe-worthy because of the similarity to Gwaihir, the king of eagles in Tolkien's works; it's like calling mages Gandalf.
Well, and thunderbirds/stormbirds are almost the same thing; of the two options, I would rather choose the one that refers to the "trio."
Secondly, the essence is more important than the appearance: the Gamayun is a prophetic bird that conveys messages to the gods. Isn't that the purpose of a magical bird? Although I don't know the lore of your birds.
I found Gveykhira in the Berserk card game. There, it was a giant reptilian bird.
Stormbirds because of the more accurate translation.
There is also the Anqa - a giant evil bird created by Allah.