So, each client counts as a distinct member of the group. Which means that you have to individually join the group from each client that you want to use. 

On the surface, this sounds annoying, but it's the same thing that you're doing with Signal or Whatsapp when you connect with the desktop app. You're just inviting your new client to the chat and then that client has it's own set of keys and state and goes forward as an independent client. 

The app itself makes sure that your messages from those distinct clients show up as from the same person. And the app itself could have the ability to sync your chat transcript between the two devices but that's not handled at the spec level here, it's handled at the application level.

Does that clarify? 

 So on the spec level there is a group of keys that are authorized to the conversation and on the app level some of the keys are the same person and need to be displayed that way? 

 Yes.