Recomputes GitHub Copilot api_mode from the selected model in the
shared /model switch path. Before this change, Copilot could carry a
stale codex_responses mode forward from a GPT-5 selection into a later
Claude model switch, causing unsupported_api_for_model errors.
Cherry-picked from #10533 by @helix4u with:
- Comment specificity (Provider-specific → Copilot api_mode override)
- Fix pre-existing duplicate opencode-go in set literal
- Extract test mock helper to reduce duplication
- Add GPT-5 → GPT-5 regression test (keeps codex_responses)
