Add subagent lineage metadata for responsesapi (#24161)

## Why

We recently added `forked_from_thread_id` which lets us trace where a
thread's _context_ comes from, but we also want to understand subagent
lineage (e.g. which parent thread spawned this subagent? what kind of
subagent is it?) which is orthogonal.

This PR adds `parent_thread_id` and `subagent_kind` to the
`x-codex-turn-metadata` header sent to ResponsesAPI.

## What changed

- Adds `parent_thread_id` and `subagent_kind` to core-owned
`x-codex-turn-metadata`.
- Restores persisted `SessionSource` and `ThreadSource` from resumed
session metadata so cold-resumed subagent threads keep their lineage on
later Responses API requests.
- Centralizes parent-thread extraction on `SessionSource` /
`SubAgentSource` and reuses it in the Responses client, analytics, agent
control, and state parsing paths.
- Extends reserved-key, git-enrichment, thread-spawn, and app-server v2
metadata coverage for the new lineage fields.

## Verification

- Not run locally per request.
- Added focused coverage in `core/src/turn_metadata_tests.rs` and
`app-server/tests/suite/v2/client_metadata.rs`.

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

@@ -1,6 +1,7 @@
 use anyhow::Result;
 use app_test_support::McpProcess;
 use app_test_support::create_fake_rollout;
+use app_test_support::create_fake_rollout_with_source;
 use app_test_support::to_response;
 use codex_app_server_protocol::JSONRPCResponse;
 use codex_app_server_protocol::RequestId;
@@ -10,6 +11,8 @@ use codex_app_server_protocol::ReviewStartResponse;
 use codex_app_server_protocol::ReviewTarget;
 use codex_app_server_protocol::ThreadForkParams;
 use codex_app_server_protocol::ThreadForkResponse;
+use codex_app_server_protocol::ThreadResumeParams;
+use codex_app_server_protocol::ThreadResumeResponse;
 use codex_app_server_protocol::ThreadSource;
 use codex_app_server_protocol::ThreadStartParams;
 use codex_app_server_protocol::ThreadStartResponse;
@@ -18,6 +21,9 @@ use codex_app_server_protocol::TurnStartResponse;
 use codex_app_server_protocol::TurnSteerParams;
 use codex_app_server_protocol::TurnSteerResponse;
 use codex_app_server_protocol::UserInput as V2UserInput;
+use codex_protocol::ThreadId as CoreThreadId;
+use codex_protocol::protocol::SessionSource;
+use codex_protocol::protocol::SubAgentSource;
 use core_test_support::responses;
 use core_test_support::skip_if_no_network;
 use pretty_assertions::assert_eq;
@@ -290,6 +296,104 @@ async fn review_start_sends_fork_lineage_in_turn_metadata_for_thread_fork_v2() -
     Ok(())
 }
 
+#[tokio::test]
+async fn turn_start_sends_subagent_lineage_after_cold_thread_resume_v2() -> Result<()> {
+    skip_if_no_network!(Ok(()));
+
+    let server = responses::start_mock_server().await;
+    let response_mock = responses::mount_sse_once(
+        &server,
+        responses::sse(vec![
+            responses::ev_response_created("resp-1"),
+            responses::ev_assistant_message("msg-1", "Done"),
+            responses::ev_completed("resp-1"),
+        ]),
+    )
+    .await;
+
+    let codex_home = TempDir::new()?;
+    create_config_toml(
+        codex_home.path(),
+        &server.uri(),
+        /*supports_websockets*/ false,
+    )?;
+
+    let parent_thread_id = CoreThreadId::new();
+    let parent_thread_id_str = parent_thread_id.to_string();
+    let subagent_thread_id = create_fake_rollout_with_source(
+        codex_home.path(),
+        "2025-01-05T12-00-00",
+        "2025-01-05T12:00:00Z",
+        "Saved subagent message",
+        Some("mock_provider"),
+        /*git_info*/ None,
+        SessionSource::SubAgent(SubAgentSource::ThreadSpawn {
+            parent_thread_id,
+            depth: 1,
+            agent_path: None,
+            agent_nickname: None,
+            agent_role: None,
+        }),
+    )?;
+
+    let mut mcp = McpProcess::new(codex_home.path()).await?;
+    timeout(DEFAULT_READ_TIMEOUT, mcp.initialize()).await??;
+
+    let resume_req = mcp
+        .send_thread_resume_request(ThreadResumeParams {
+            thread_id: subagent_thread_id.clone(),
+            ..Default::default()
+        })
+        .await?;
+    let resume_resp: JSONRPCResponse = timeout(
+        DEFAULT_READ_TIMEOUT,
+        mcp.read_stream_until_response_message(RequestId::Integer(resume_req)),
+    )
+    .await??;
+    let ThreadResumeResponse { thread, .. } = to_response::<ThreadResumeResponse>(resume_resp)?;
+    assert_eq!(thread.id, subagent_thread_id);
+
+    let turn_req = mcp
+        .send_turn_start_request(TurnStartParams {
+            thread_id: thread.id.clone(),
+            input: vec![V2UserInput::Text {
+                text: "Continue".to_string(),
+                text_elements: Vec::new(),
+            }],
+            ..Default::default()
+        })
+        .await?;
+    let turn_resp: JSONRPCResponse = timeout(
+        DEFAULT_READ_TIMEOUT,
+        mcp.read_stream_until_response_message(RequestId::Integer(turn_req)),
+    )
+    .await??;
+    let TurnStartResponse { turn } = to_response::<TurnStartResponse>(turn_resp)?;
+
+    timeout(
+        DEFAULT_READ_TIMEOUT,
+        mcp.read_stream_until_notification_message("turn/completed"),
+    )
+    .await??;
+
+    let request = response_mock.single_request();
+    let metadata = request
+        .header("x-codex-turn-metadata")
+        .as_deref()
+        .map(parse_json_header)
+        .unwrap_or_else(|| panic!("missing x-codex-turn-metadata header"));
+    assert_eq!(
+        metadata["parent_thread_id"].as_str(),
+        Some(parent_thread_id_str.as_str())
+    );
+    assert_eq!(metadata["subagent_kind"].as_str(), Some("thread_spawn"));
+    assert_eq!(metadata["thread_id"].as_str(), Some(thread.id.as_str()));
+    assert_eq!(metadata["turn_id"].as_str(), Some(turn.id.as_str()));
+    assert!(metadata.get("forked_from_thread_id").is_none());
+
+    Ok(())
+}
+
 #[tokio::test]
 async fn turn_steer_updates_client_metadata_on_follow_up_responses_request_v2() -> Result<()> {
     skip_if_no_network!(Ok(()));