Here I try to sumarise all the possible options of this one/many, body/mind product:
- Independent minds: Many fully conscious individuals (one mind per individual), mutually disconnected.
- Networked/Connected minds: Many fully conscious individuals connected in a telephatic network.
- Hive-mind: many unconscious individuals that forms a single superior entity through their connection.
- Over-mind: many unconscious individuals controlled by a single fully conscious individual (the "brain").
Independent is the case to which the others are compared. Think of Humans. We can be corrupted, we can rebel and attack each other, our governments can fool us...
Networked implies that each body is a fully autonomous individual (as in Independent) that can connect to the network of minds of the people of the same species, retaining its individuality (just improving communications, mutual understanding, social organisation, etc.). Laenfa or Trith could be of this kind (have this trait), since they are telephatic. However, this connection is not necessarilly via telephathy (the special trait to read/write in other's minds), it can be via a complex, extracorporeal, robotic/chemical communications system natural and specific to the species. So being a Networked species does not imply having telephathy, and maybe Gysache could be given this trait (from being a very herdy and interconnected for own survival). Also, the members of Networked species are not mental slaves of any ubermind and disagreements can arise between parts of the networked mind (e.g. the individuals of a colony can get angry at their fellows for letting them under enemy siege for long and rebel).
Hive-mind is a step forward towards less autonomous individuals and more cohesive mind. Each physical individual has a rudimentary functional mind, and like the previous case, it connects to a network of minds. But in this case each individual losses its individuality (or probably has little to none from start, mostly based on automatisms/instincts) and the networks itself becomes an individual on its own. We can think of ants, for which each ant is a simple creature but in combination with many others (of the same nest) they behave as a distributed individual capable of superior things from collective intelligence (the ant queen does not control the hive, she is one more of the simple minds that form the hive-mind, just with singular body capabilities). So there is no Capital for this kind of species. George would be this kind.
Over-mind is similar to Hive-mind in that you have many simple individuals and a superior mind, but in this case the superior mind is not the result of a collective intelligence but it's contained within a single, overpowered thinking being (the "brain") that controls all the other minions. This could be the Zergs of Star Craft and the insectoids of Ender's Game. Depending on other characteristics of the species, this could imply that killing the "capital" completely kills the empire if the brain can't scape, or that it will initiate a process of growing a new brain somewhere else while the entire "body" (the rest of "unbrained" specimens) of the empire is convulsing, maybe several brains will start to grow and compete against each other to be the final boss even causing fragmentation of the empire (AI takes control), this new-brain-growing process could be something for which the empire is prepared in advance and the bad effects are mild and short-termed, or the opposite. So here we could have one special negative trait only for over-mind species, “irreplaceable brain”, which makes your empire automatically and completely lose if your capital is conquered.
I've discarded the few categories that can be thought of by blindly combining the possible concepts, like "multiple minds in a single body" (that makes no sense or equals type Independent in capabilities), and "single mind in a single-yet-distributed body" which effectively is the same as Hive-mind. But I don't discard having left out something important or interesting.
I'm working on a proposal for influence-related species traits. So any thoughts about this mind stuff you may want to share will be very welcome.