feat(analytics): generate an installation_id and pass it in responsesapi client_metadata (#16912)

## Summary

This adds a stable Codex installation ID and includes it on Responses
API requests via `x-codex-installation-id` passed in via the
`client_metadata` field for analytics/debugging.

The main pieces are:
- persist a UUID in `$CODEX_HOME/installation_id`
- thread the installation ID into `ModelClient`
- send it in `client_metadata` on Responses requests so it works
consistently across HTTP and WebSocket transports

codex-rs/core/tests/suite/client.rs

@@ -81,6 +81,8 @@ use wiremock::matchers::method;
 use wiremock::matchers::path;
 use wiremock::matchers::query_param;
 
+const INSTALLATION_ID_FILENAME: &str = "installation_id";
+
 #[expect(clippy::unwrap_used)]
 fn assert_message_role(request_body: &serde_json::Value, role: &str) {
     assert_eq!(request_body["role"].as_str().unwrap(), role);
@@ -760,10 +762,18 @@ async fn includes_conversation_id_and_model_headers_in_request() {
         .header("authorization")
         .expect("authorization header");
     let request_originator = request.header("originator").expect("originator header");
+    let request_body = request.body_json();
+    let installation_id =
+        std::fs::read_to_string(test.codex_home_path().join(INSTALLATION_ID_FILENAME))
+            .expect("read installation id");
 
     assert_eq!(request_session_id, session_id.to_string());
     assert_eq!(request_originator, originator().value);
     assert_eq!(request_authorization, "Bearer Test API Key");
+    assert_eq!(
+        request_body["client_metadata"]["x-codex-installation-id"].as_str(),
+        Some(installation_id.as_str())
+    );
 }
 
 #[tokio::test(flavor = "multi_thread", worker_threads = 2)]
@@ -868,6 +878,7 @@ async fn send_provider_auth_request(server: &MockServer, auth: ModelProviderAuth
             "unused-api-key",
         ))),
         conversation_id,
+        /*installation_id*/ "11111111-1111-4111-8111-111111111111".to_string(),
         provider,
         SessionSource::Exec,
         config.model_verbosity,
@@ -1048,11 +1059,18 @@ async fn chatgpt_auth_sends_correct_request() {
     let request_body = request.body_json();
 
     let session_id = request.header("session_id").expect("session_id header");
+    let installation_id =
+        std::fs::read_to_string(test.codex_home_path().join(INSTALLATION_ID_FILENAME))
+            .expect("read installation id");
     assert_eq!(session_id, thread_id.to_string());
 
     assert_eq!(request_originator, originator().value);
     assert_eq!(request_authorization, "Bearer Access Token");
     assert_eq!(request_chatgpt_account_id, "account_id");
+    assert_eq!(
+        request_body["client_metadata"]["x-codex-installation-id"].as_str(),
+        Some(installation_id.as_str())
+    );
     assert!(request_body["stream"].as_bool().unwrap());
     assert_eq!(
         request_body["include"][0].as_str().unwrap(),
@@ -2179,6 +2197,7 @@ async fn azure_responses_request_includes_store_and_reasoning_ids() {
     let client = ModelClient::new(
         /*auth_manager*/ None,
         conversation_id,
+        /*installation_id*/ "11111111-1111-4111-8111-111111111111".to_string(),
         provider.clone(),
         SessionSource::Exec,
         config.model_verbosity,

codex-rs/core/tests/suite/client_websockets.rs

@@ -55,6 +55,7 @@ const MODEL: &str = "gpt-5.2-codex";
 const OPENAI_BETA_HEADER: &str = "OpenAI-Beta";
 const WS_V2_BETA_HEADER_VALUE: &str = "responses_websockets=2026-02-06";
 const X_CLIENT_REQUEST_ID_HEADER: &str = "x-client-request-id";
+const TEST_INSTALLATION_ID: &str = "11111111-1111-4111-8111-111111111111";
 
 fn assert_request_trace_matches(body: &serde_json::Value, expected_trace: &W3cTraceContext) {
     let client_metadata = body["client_metadata"]
@@ -125,6 +126,10 @@ async fn responses_websocket_streams_request() {
         handshake.header(X_CLIENT_REQUEST_ID_HEADER),
         Some(harness.conversation_id.to_string())
     );
+    assert_eq!(
+        body["client_metadata"]["x-codex-installation-id"].as_str(),
+        Some(TEST_INSTALLATION_ID)
+    );
 
     server.shutdown().await;
 }
@@ -1756,6 +1761,7 @@ async fn websocket_harness_with_provider_options(
     let client = ModelClient::new(
         /*auth_manager*/ None,
         conversation_id,
+        /*installation_id*/ TEST_INSTALLATION_ID.to_string(),
         provider.clone(),
         SessionSource::Exec,
         config.model_verbosity,