fix(app-server): emit turn/started only when turn actually starts (#13261)

This is a follow-up for https://github.com/openai/codex/pull/13047

## Why
We had a race where `turn/started` could be observed before the thread
had actually transitioned to `Active`. This was because we eagerly
emitted `turn/started` in the request handler for `turn/start` (and
`review/start`).

That was showing up as flaky `thread/resume` tests, but the real issue
was broader: a client could see `turn/started` and still get back an
idle thread immediately afterward.

The first idea was to eagerly call
`thread_watch_manager.note_turn_started(...)` from the `turn/start`
request path. That turns out to be unsafe, because
`submit(Op::UserInput)` only queues work. If a turn starts and completes
quickly, request-path bookkeeping can race with the real lifecycle
events and leave stale running state behind.

**The real fix** is to move `turn/started` to emit only after the turn
_actually_ starts, so we do that by waiting for the
`EventMsg::TurnStarted` notification emitted by codex core. We do this
for both `turn/start` and `review/start`.

I also verified this change is safe for our first-party codex apps -
they don't have any assumptions that `turn/started` is emitted before
the RPC response to `turn/start` (which is correct anyway).

I also removed `single_client_mode` since it isn't really necessary now.

## Testing
- `cargo test -p codex-app-server thread_resume -- --nocapture`
- `cargo test -p codex-app-server
'suite::v2::turn_start::turn_start_emits_notifications_and_accepts_model_override'
-- --exact --nocapture`
- `cargo test -p codex-app-server`

codex-rs/app-server/tests/suite/v2/thread_resume.rs

@@ -1,4 +1,3 @@
-use anyhow::Context;
 use anyhow::Result;
 use app_test_support::McpProcess;
 use app_test_support::create_apply_patch_sse_response;
@@ -17,7 +16,6 @@ use codex_app_server_protocol::FileChangeApprovalDecision;
 use codex_app_server_protocol::FileChangeRequestApprovalResponse;
 use codex_app_server_protocol::ItemStartedNotification;
 use codex_app_server_protocol::JSONRPCError;
-use codex_app_server_protocol::JSONRPCNotification;
 use codex_app_server_protocol::JSONRPCResponse;
 use codex_app_server_protocol::PatchApplyStatus;
 use codex_app_server_protocol::PatchChangeKind;
@@ -30,7 +28,6 @@ use codex_app_server_protocol::ThreadResumeResponse;
 use codex_app_server_protocol::ThreadStartParams;
 use codex_app_server_protocol::ThreadStartResponse;
 use codex_app_server_protocol::ThreadStatus;
-use codex_app_server_protocol::ThreadStatusChangedNotification;
 use codex_app_server_protocol::TurnStartParams;
 use codex_app_server_protocol::TurnStartResponse;
 use codex_app_server_protocol::TurnStatus;
@@ -293,7 +290,7 @@ async fn thread_resume_keeps_in_flight_turn_streaming() -> Result<()> {
     .await??;
     timeout(
         DEFAULT_READ_TIMEOUT,
-        wait_for_thread_status_active(&mut primary, &thread.id),
+        primary.read_stream_until_notification_message("turn/started"),
     )
     .await??;
 
@@ -400,7 +397,7 @@ async fn thread_resume_rejects_history_when_thread_is_running() -> Result<()> {
         to_response::<TurnStartResponse>(running_turn_resp)?;
     timeout(
         DEFAULT_READ_TIMEOUT,
-        wait_for_thread_status_active(&mut primary, &thread_id),
+        primary.read_stream_until_notification_message("turn/started"),
     )
     .await??;
 
@@ -516,7 +513,7 @@ async fn thread_resume_rejects_mismatched_path_when_thread_is_running() -> Resul
         to_response::<TurnStartResponse>(running_turn_resp)?;
     timeout(
         DEFAULT_READ_TIMEOUT,
-        wait_for_thread_status_active(&mut primary, &thread_id),
+        primary.read_stream_until_notification_message("turn/started"),
     )
     .await??;
 
@@ -619,7 +616,7 @@ async fn thread_resume_rejoins_running_thread_even_with_override_mismatch() -> R
     .await??;
     timeout(
         DEFAULT_READ_TIMEOUT,
-        wait_for_thread_status_active(&mut primary, &thread.id),
+        primary.read_stream_until_notification_message("turn/started"),
     )
     .await??;
 
@@ -1419,30 +1416,6 @@ required = true
     )
 }
 
-async fn wait_for_thread_status_active(
-    mcp: &mut McpProcess,
-    thread_id: &str,
-) -> Result<ThreadStatusChangedNotification> {
-    loop {
-        let status_changed_notif: JSONRPCNotification = mcp
-            .read_stream_until_notification_message("thread/status/changed")
-            .await?;
-        let status_changed_params = status_changed_notif
-            .params
-            .context("thread/status/changed params must be present")?;
-        let status_changed: ThreadStatusChangedNotification =
-            serde_json::from_value(status_changed_params)?;
-        if status_changed.thread_id == thread_id
-            && status_changed.status
-                == (ThreadStatus::Active {
-                    active_flags: Vec::new(),
-                })
-        {
-            return Ok(status_changed);
-        }
-    }
-}
-
 #[allow(dead_code)]
 fn set_rollout_mtime(path: &Path, updated_at_rfc3339: &str) -> Result<()> {
     let parsed = chrono::DateTime::parse_from_rfc3339(updated_at_rfc3339)?.with_timezone(&Utc);