I think your definition is too restrictive, and inapplicable. I doubt we can really say enough about how planets formed, to be able to use that as a definition. For example, suppose you found a rocky body out in the Kuiper belt, in an eccentric orbit? Would you propose that it had formed in the proplyd and got ejected, and so was a planet, while its neighbours were not?

I am more inclined to call just about anything whose shortest orbital period is around a star, a planet, and to accept a range of subtypes. Gas giants, rocks, snowballs, Trojans,...