The manager that records each turn's event, the run's transcript store and the run the sink writes.
Error when the transcript store does not implement appendNext.
Retains the numbered-turn entry point for existing callers.
Stores the final as the run's next turn and records its bot:transcript event; undefined when a fence, a run that is over or a run that takes no turns yet (MeetingManager.takesTurns) drops it.
Turn ids are assigned by the sink in arrival order, using the store's atomic counter for the run across sink instances.