Basically, kaki and American persimmons are too far apart genetically to naturally create hybrids. Only a couple of “original” hybrids have been created under very controlled circumstances and they are generally attributed to “embyro rescue” of only partially formed/matured seeds. I believe the only hybrids created this way were Rosseyanka and JT-02, although there might be others that never really got into circulation if they weren’t good enough or didn’t survive. Everything else comes from those, primarily from Rosseyanka at this point, with there now being 3rd and 4th (maybe even 5th) generation crosses from those.
Sometimes people will cross two hybrids, but often it seems we get crosses from a hybrid crossed to a kaki or American persimmon seeking to get the size of kaki or the taste of American. The ability to create viable seed varies by hybrid, often relating to how close the parent is in terms of kaki/American balance, so it is hard to get viable seed from a “mostly” kaki hybrid crossed with an American persimmon male for instance.
For a long trip down the rabbit hole with the prize of an excellent list complied by @Richard you can go here: