Perhaps you could lean into the trunk of a tree concept, by having the difeerentiation being a species of tree that exists only on one of each of the six worlds leading your conglomerate?

Or maybe a different name for the same species? So maybe on one world Giant Seqouia's are called "High Tops", on another they are called "Redwood", on another they are called "Smooth Trunk", on another they are called "Small Cone"...

Just an example, and picking better names would be good, but maybe it's one way of diferentiating? And then of course, it becomes the convention to refer to the G6 worlds by their "Tree Name"

Then other worlds could be their tree name with a number indicating the order that G6 world "colonised" them.

Then, Earth. If you are positing it is NOT colonised, they could name it in their system, but the Earthlings could reject this and call themselves by something entirely different, Like Solarians...