That's more of a definition, that what I've been getting. To say its a host who shares the same essence as the Phoenix, makes pretty good sense of that term.
I'm trying to work with a fan collab Marvel Anime writer, and I wanted to make sure that, her new definitions of "One True Phoenix" doesn't stray too far from canon.
In this animated series, the title One True Phoenix belongs to two individuals and has two different definitions, due to how her story plays out. One True Phoenix in relation to Jean Grey, has to do with the fact the Phoenix had lost a part of itself, and after being able to perfectly expand all its powers within her, she became recognized as the "One True Phoenix".
The other revolves around another X-Man, and he becomes the "One True Phoenix" for a very different reason. He winds up bonding with a new Cosmic Entity, and... because he is somehow connected this mysterious Cosmic Origin, he is the one and only host that is the... genetic human incarnation or avatar, for said Cosmic Being. Like... the animator said a lot of metaphysical stuff with the singularity science and mathematics- she's awesome and very intelligent, but I can't recite what she said about him being the One True Phoenix.