Record one in-call chat message so it survives the room.

This is not the delivery path. Messages reach the other participants over the LiveKit data channel; the client calls this alongside that, after it has already published. Persistence must never be able to delay or drop a message in a live class, so a failure here costs the record and not the conversation — which is why the client fires it and does not await it.

Idempotent on client_id: a retry, or the class broadcast echoing a message back into its own room, updates nothing and returns the existing row.