Send a message to a chat session

Send a user message to a chat session (REST only, no streaming).

Delegates to :func:deliver_user_message_to_session, the canonical non-WebSocket send path also used by board-card "send to agent" and gate-answer flows. That helper persists the user message and drives the orchestrator: steering an active run (mid-run queue) or starting a fresh turn (idle → dispatch). The persisted user message is fetched back by its client_msg_id and returned so the client can render the bubble immediately.

A delivery status of "locked" (another run holds the orchestrator lock) surfaces as a 409 session_locked; "busy" (the mid-run queue rejected the enqueue) surfaces as a 409 session_busy. In both cases the message may still have been persisted, but the orchestrator did not accept it for a turn, so the client should retry.