Distinguish turn-start thread persistence (#37926)

## What changed

- Add `PersistContext` to the thread-store persistence contract so stores can
  identify persistence requested immediately before model sampling. A
  `TurnStart` request may be enqueued in the background when later flush and
  shutdown operations fence it and report any failure; all other persistence
  remains synchronous.
- Use the turn-start context only for the initial input of a regular turn, while
  admission acknowledgments, steered input, and other persistence paths keep
  the standard durability behavior.
- Flush pending metadata before turn-start persistence and ensure shutdown is
  still attempted if that metadata update fails.

## Testing

- Verify that developer instructions and user input are persisted before the
  model request completes for a default paginated-history thread.

GitOrigin-RevId: 162f0ec796a61aebe66ca0b909fbcccec3047b85

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

@@ -1431,7 +1431,8 @@ impl Session {
             }
             if matches!(&sess.fork_persistence, ForkPersistence::Referenced { .. }) {
                 // Keep the source reserved until the child's history reference is durable.
-                sess.try_ensure_rollout_materialized().await?;
+                sess.try_ensure_rollout_materialized(PersistContext::Standard)
+                    .await?;
             }
             {
                 let mut state = sess.state.lock().await;