Pi engine: route OpenAI/Codex backend through the Responses API (#56692)

* Initial plan

* Route Pi's OpenAI/Codex backend through the Responses API

Co-authored-by: pelikhan <4175913+pelikhan@users.noreply.github.com>

* Address code review feedback: clarify anthropic gateway api comment/test naming

Co-authored-by: pelikhan <4175913+pelikhan@users.noreply.github.com>

* Fix Codex Pi gateway API routing

Co-authored-by: gh-aw-bot <259018956+gh-aw-bot@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
Co-authored-by: pelikhan <4175913+pelikhan@users.noreply.github.com>
Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>
Co-authored-by: gh-aw-bot <259018956+gh-aw-bot@users.noreply.github.com>