Allow background persistence for steered user input (#45506)

## Why

Persisting user input received during an active turn currently blocks the next model request. Stores that support background persistence can overlap this checkpoint with inference.

## What changed

- Add `PersistContext::SteeredUserInput` and `allows_background_persistence()` so stores may enqueue these checkpoints, with durability and error reporting enforced by later flush or shutdown operations.
- Use the new context for accepted steered user input and apply the same metadata handling as turn-start persistence.
- Keep tool outputs synchronous, including in mixed input batches, and allow stores to retain synchronous persistence for all contexts.

## Testing

Add gated-store integration tests covering background user-input persistence, synchronous stores, and synchronous tool-output checkpoints. Verify that the next request includes the steered input and waits for persistence when required.

GitOrigin-RevId: c60b7b6c9b483245fd3169306bcf0de248ccdf35

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

@@ -438,7 +438,7 @@ pub(crate) async fn run_turn(
             &turn_context,
             &turn_context.capture_current_model_info(),
             &pending_input,
-            PersistContext::Standard,
+            PersistContext::SteeredUserInput,
         )
         .await
         {
@@ -834,13 +834,21 @@ pub(crate) async fn run_hooks_and_record_inputs(
             if matches!(input_item, TurnInput::UserInput { content, .. } if !content.is_empty()) {
                 accepted_user_input = true;
             }
+            // Tool outputs retain their durability barrier, including in mixed input batches.
+            let input_persist_context = if persist_context == PersistContext::SteeredUserInput
+                && matches!(input_item, TurnInput::FunctionCallOutput(_))
+            {
+                PersistContext::Standard
+            } else {
+                persist_context
+            };
             record_pending_input(
                 sess,
                 turn_context,
                 model_info,
                 input_item.clone(),
                 hook_outcome.additional_contexts,
-                persist_context,
+                input_persist_context,
             )
             .await;
         }