Which came first, the phoenix or the flame?

Like a tongue on frozen steel, like flesh in flame —

Fawkes is a phoenix, Harry. Phoenixes burst into flame when it is time for them to die and are reborn from the ashes.

Then he heard a terrible cry that pulled at his insides, that expressed agony of a kind that neither flame nor curse could cause, and he stood up, swaying, more frightened than he had been that day, more frightened, perhaps, than he had been in his whole life...