Preserve turn triggers across delegated agent work (#44659)

## Why

Delegated usage should retain the trigger of the turn that initiated it, including composer input and scheduled automation.

## What changed

Propagate `turn_trigger` through agent spawning and follow-up messages in both multi-agent tool implementations. Apply the supplied trigger when pending mailbox work starts a new turn.

## Testing

Extend tests to verify trigger propagation to spawned and nested agents, updated triggers on reused agents and peer follow-ups, and preservation of the active turn's trigger when steering.

GitOrigin-RevId: e8fea208587c633fec7cb3f1c0815ff91d1a44e5

codex-rs/core/src/session/turn_input_tests.rs

@@ -858,6 +858,9 @@ async fn steer_only_requires_active_turn() {
 #[tokio::test]
 async fn steer_only_enforces_expected_turn_id() {
     let (session, turn_context, _rx) = make_session_and_context_with_rx().await;
+    turn_context
+        .turn_metadata_state
+        .set_turn_trigger("composer".to_string());
     session
         .spawn_task(
             Arc::clone(&turn_context),
@@ -904,7 +907,12 @@ async fn steer_only_enforces_expected_turn_id() {
 
     let submission = handle(
         &session,
-        TurnInputRequest::new(SubmittedTurnInput::ResponseItem(output)),
+        TurnInputRequest::new(SubmittedTurnInput::ResponseItem(output)).on_start(
+            TurnStartOptions {
+                turn_trigger: Some("automation_cron_scheduled".to_string()),
+                ..Default::default()
+            },
+        ),
         TurnInputMode::StartOrSteer,
         "test-submission".to_string(),
     )
@@ -917,6 +925,13 @@ async fn steer_only_enforces_expected_turn_id() {
             turn_id: turn_context.sub_id.clone()
         }
     );
+    assert_eq!(
+        turn_context
+            .turn_metadata_state
+            .current_turn_trigger()
+            .as_deref(),
+        Some("composer")
+    );
     let turn_state = session
         .input_queue
         .turn_state_for_sub_id(&session.active_turn, &turn_context.sub_id)