[Empathy] is the foreground version of the shell's Message Tray chat.
When the Empathy window for a specific conversation is in the foreground we should not see the background Tray version.
However, when I don't see the conversation window I expect to get notifications for updates to that (ongoing) conversation.
These messages should remain in the Tray until I see them and then for a certain amount of time after that (to be determined) at which point we deem the conversation to have lapsed. Unless the conversation is pinned to the Tray.
The difference between Music and Chat here is, as mentioned above, that Chat is a service. We don't sign the user out when the chat app/window is closed. Similarly we shouldn't deem the current ongoing conversation to be finished. Especially since that isn't indicated or expected on the other end of the conversation.
That said we do need a way to ignore further updates to a conversation. My mom is really a pain sometimes. We often disagree on when a chat is over. It would be best to have a way to not be notified of updates to the conversation. Again "conversation" being a time limited scope.
So, basically closing the chat app window should move all further interaction to the Tray until the conversation is complete.