## 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`
## Summary
Responses Lite does not execute hosted Responses tools, so models using
it must route web search and image generation through Codex-owned
executors & standalone Response's API endpoints.
This PR is stacked on #26487.
## Validation
- `cargo test -p codex-core responses_lite_ --lib`
- `cargo test -p codex-core
standalone_executors_remain_hidden_without_flags_or_responses_lite
--lib`
- `cargo test -p codex-core
hosted_tools_follow_provider_auth_model_and_config_gates --lib`
- `cargo test -p codex-web-search-extension -p
codex-image-generation-extension`
- `cargo test -p codex-app-server --test all standalone_`
- `cargo fmt --all -- --check`
rka-oai