Okay, this bug has been fixed!
I’m embarrassed by such a glaring bug. 
Thank you for the report and your patience. Unfortunately, we won’t be able to recover your lost chat because it was never saved.
The reason for the bug, if you’re curious, is that the chat editor considers notebooks to be “read-only” because they can only be edited through chat messages; normal cell editing affordances are disabled. And I recently changed the notebook editor so that changes aren’t saved when the notebook is read-only (fixing a different bug related to programmatic file attachments and the republish button, reported here), forgetting that this would also prevent chats from being saved.
I’m now implementing persistence for Agent conversations in notebooks, so you should soon be able to save your in-notebook chats there, too. Very sorry, again!