I don't think that the first people of either nation (except for orcs) had belly buttons (considering that you get one only after you are born - ie when the umbilical cord is cut and made into a knot). The Elves at Cuivienen weren't born - they just awoke. Same goes for Dwarves and men. Orcs are different, since they are "made" from pre-existing Elves, that might have had belly buttons.
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You passed from under darkened dome, you enter now the secret land. - Take me to Finrod's fabled home!... ~ Finrod: The Rock Opera
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