Support alternative websocket API (#10861)

**Test plan**

```
cargo build -p codex-cli && RUST_LOG='codex_api::endpoint::responses_websocket=trace,codex_core::client=debug,codex_core::codex=debug' \
  ./target/debug/codex \
    --enable responses_websockets_v2 \
    --profile byok \
    --full-auto
```

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

@@ -1273,6 +1273,7 @@ async fn azure_responses_request_includes_store_and_reasoning_ids() {
         false,
         false,
         false,
+        false,
         None,
     );
     let mut client_session = client.new_session();