[codex] Send Responses Lite transport header (#26542)

## Summary

- send `X-OpenAI-Internal-Codex-Responses-Lite: true` on HTTP Responses
requests and WebSocket upgrade requests when model metadata enables
Responses Lite
- use client metadata when sending it over the websocket

This PR is stacked on #26490.

## Why

The Responses Lite marker is request-scoped for HTTP but
connection-scoped for Responses-over-WebSocket because it is carried on
the upgrade request. Reusing a cached socket opened for the opposite
mode would therefore send the wrong transport contract.

## Validation

- `just test -p codex-core responses_lite`
- `just test -p codex-core
responses_websocket_reconnects_when_responses_lite_mode_changes`
- `just fix -p codex-core`
- `just fmt`

codex-rs/core/src/client_tests.rs

@@ -293,7 +293,10 @@ fn build_ws_client_metadata_includes_window_lineage_and_turn_metadata() {
 
     client.advance_window_generation();
 
-    let client_metadata = client.build_ws_client_metadata(Some(r#"{"turn_id":"turn-123"}"#));
+    let client_metadata = client.build_ws_client_metadata(
+        Some(r#"{"turn_id":"turn-123"}"#),
+        /*use_responses_lite*/ false,
+    );
     let thread_id = client.state.thread_id;
     assert_eq!(
         client_metadata,