Add turn start timestamp to turn metadata (#19473)

## Why
- Without change: MCP tool calls receive
`_meta["x-codex-turn-metadata"]` with `session_id` and `turn_id`.
- Issue: MCP servers may want the turn start timestamp to measure
internal latency relative to turn start.

## What Changed
- With change: turn metadata now includes `turn_started_at_unix_ms`,
which is propagated to MCP tool calls in
`_meta["x-codex-turn-metadata"]`.

## Verification
- `codex-rs/core/src/mcp_tool_call_tests.rs`
- `codex-rs/core/src/turn_metadata_tests.rs`
- `codex-rs/core/src/turn_timing_tests.rs`
- `codex-rs/core/tests/responses_headers.rs`
- `codex-rs/core/tests/suite/search_tool.rs`

codex-rs/core/src/turn_timing_tests.rs

@@ -5,6 +5,8 @@ use codex_protocol::models::FunctionCallOutputPayload;
 use codex_protocol::models::ResponseItem;
 use pretty_assertions::assert_eq;
 use std::time::Instant;
+use std::time::SystemTime;
+use std::time::UNIX_EPOCH;
 
 use super::TurnTimingState;
 use super::response_item_records_turn_ttft;
@@ -76,6 +78,27 @@ async fn turn_timing_state_records_ttfm_independently_of_ttft() {
     );
 }
 
+#[tokio::test]
+async fn turn_timing_state_records_turn_started_epoch_millis() {
+    let state = TurnTimingState::default();
+    let before = SystemTime::now()
+        .duration_since(UNIX_EPOCH)
+        .expect("system time should be after unix epoch")
+        .as_millis();
+
+    let started_at_unix_ms = state.mark_turn_started(Instant::now()).await;
+
+    let after = SystemTime::now()
+        .duration_since(UNIX_EPOCH)
+        .expect("system time should be after unix epoch")
+        .as_millis();
+    assert!(u128::try_from(started_at_unix_ms).is_ok_and(|ms| before <= ms && ms <= after));
+    assert_eq!(
+        state.started_at_unix_secs().await,
+        Some(started_at_unix_ms / 1000)
+    );
+}
+
 #[test]
 fn response_item_records_turn_ttft_for_first_output_signals() {
     assert!(response_item_records_turn_ttft(