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/codex-api/src/common.rs

@@ -169,6 +169,8 @@ pub struct ResponsesApiRequest {
     pub prompt_cache_key: Option<String>,
     #[serde(skip_serializing_if = "Option::is_none")]
     pub text: Option<TextControls>,
+    #[serde(skip_serializing_if = "Option::is_none")]
+    pub client_metadata: Option<HashMap<String, String>>,
 }
 
 impl From<&ResponsesApiRequest> for ResponseCreateWsRequest {
@@ -189,7 +191,7 @@ impl From<&ResponsesApiRequest> for ResponseCreateWsRequest {
             prompt_cache_key: request.prompt_cache_key.clone(),
             text: request.text.clone(),
             generate: None,
-            client_metadata: None,
+            client_metadata: request.client_metadata.clone(),
         }
     }
 }

codex-rs/codex-api/tests/clients.rs

@@ -278,6 +278,7 @@ async fn streaming_client_retries_on_transport_error() -> Result<()> {
         service_tier: None,
         prompt_cache_key: None,
         text: None,
+        client_metadata: None,
     };
     let client = ResponsesClient::new(transport.clone(), provider, NoAuth);
 
@@ -320,6 +321,7 @@ async fn azure_default_store_attaches_ids_and_headers() -> Result<()> {
         service_tier: None,
         prompt_cache_key: None,
         text: None,
+        client_metadata: None,
     };
 
     let mut extra_headers = HeaderMap::new();