[codex] Omit fork turns from thread started notifications (#19093)

## Why

`thread/fork` responses intentionally include copied history so the
caller can render the fork immediately, but `thread/started` is a
lifecycle notification. The v2 `Thread` contract says notifications
should return `turns: []`, and the fork path was reusing the response
thread directly, causing copied turns to be emitted through
`thread/started` as well.

## What Changed

- Route app-server `thread/started` notification construction through a
helper that clears `thread.turns` before sending.
- Keep `thread/fork` responses unchanged so callers still receive copied
history.
- Add persistent and ephemeral fork coverage that asserts
`thread/started` emits an empty `turns` array while the response retains
fork history.

## Testing

- `just fmt`
- `cargo test -p codex-app-server`

codex-rs/app-server/src/codex_message_processor.rs

@@ -2859,7 +2859,7 @@ impl CodexMessageProcessor {
                     ))
                     .await;
 
-                let notif = ThreadStartedNotification { thread };
+                let notif = thread_started_notification(thread);
                 listener_task_context
                     .outgoing
                     .send_server_notification(ServerNotification::ThreadStarted(notif))
@@ -5365,7 +5365,7 @@ impl CodexMessageProcessor {
             .await;
         }
 
-        let notif = ThreadStartedNotification { thread };
+        let notif = thread_started_notification(thread);
         self.outgoing
             .send_server_notification(ServerNotification::ThreadStarted(notif))
             .await;
@@ -7744,7 +7744,7 @@ impl CodexMessageProcessor {
                             .await,
                         /*has_in_progress_turn*/ false,
                     );
-                    let notif = ThreadStartedNotification { thread };
+                    let notif = thread_started_notification(thread);
                     self.outgoing
                         .send_server_notification(ServerNotification::ThreadStarted(notif))
                         .await;
@@ -10094,6 +10094,11 @@ fn build_thread_from_snapshot(
     }
 }
 
+fn thread_started_notification(mut thread: Thread) -> ThreadStartedNotification {
+    thread.turns.clear();
+    ThreadStartedNotification { thread }
+}
+
 pub(crate) fn summary_to_thread(
     summary: ConversationSummary,
     fallback_cwd: &AbsolutePathBuf,