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/app-server/tests/suite/v2/thread_read.rs

@@ -76,6 +76,7 @@ use codex_thread_store::CreateThreadParams;
 use codex_thread_store::InMemoryThreadStore;
 use codex_thread_store::LocalThreadStore;
 use codex_thread_store::LocalThreadStoreConfig;
+use codex_thread_store::PersistContext;
 use codex_thread_store::ThreadMetadataPatch;
 use codex_thread_store::ThreadPersistenceMetadata;
 use codex_thread_store::ThreadStore;
@@ -487,7 +488,9 @@ async fn thread_search_occurrences_reads_paginated_projection() -> Result<()> {
             },
         })
         .await?;
-    store.persist_thread(thread_id).await?;
+    store
+        .persist_thread(thread_id, PersistContext::Standard)
+        .await?;
     store
         .append_items(AppendThreadItemsParams {
             thread_id,
@@ -1535,7 +1538,9 @@ async fn paginated_history_lists_and_legacy_reads_use_projected_turns_and_items(
             },
         })
         .await?;
-    store.persist_thread(thread_id).await?;
+    store
+        .persist_thread(thread_id, PersistContext::Standard)
+        .await?;
     store
         .append_items(AppendThreadItemsParams {
             thread_id,