I don't think any of those examples negate that the Phoenix is a part of Jean. I like to akin it to the idea of a soulmate. We know from Claremont that part of Jean's power is rebirth. For all we know that part of her could very well be Phoenix. Aside from Rachel though the Phoenix has inhabited many, non of those have been successful and that could be most arguably attributed as you stated Jean Grey's genetics. What we do know is that though out all of it's appearances the Phoenix has preferred jean over them all, including Rachel as she has stated on several occasions and even called out to her mother when she lost her phoenix abilities during Rise of the Shiar. So to me Phoenix is a part of Jean's soul and is ok travelling around in the bodies of compatible or semi-compatable bodies and enjoy life while burning them out until it's one true body, Jean's appear. JMO though.