Problem

After Responses API support was added in #5783, OpenAI provider routing is still inconsistent:

1. gpt-5.2-codex can be routed via chat/completions instead of responses.
2. OPENAI_BASE_PATH handling is inconsistent across request types (chat/completions vs responses vs models).
3. Endpoint selection relies on brittle hardcoded checks and string replacement assumptions.

This causes real failures for codex models and confusing behavior when users configure custom OpenAI-compatible base paths.

Reproduction

1. Configure provider openai with model gpt-5.2-codex and default OPENAI_BASE_PATH=v1/chat/completions.
2. Run a normal completion/streaming request.
3. Observe request routed incorrectly for some codex variants, or inconsistent endpoint handling depending on path config.

Additional repro for custom paths:

1. Set OPENAI_HOST to a proxy host.
2. Set OPENAI_BASE_PATH=openai/v1/chat/completions.
3. Fetch models / run streaming / run non-streaming.
4. Observe path handling differences between responses and models endpoints.

Expected Behavior

• gpt-5.*-codex should default to Responses API routing when no explicit custom override is provided.
• If OPENAI_BASE_PATH explicitly points to .../chat/completions, routing should remain chat-completions for compatibility.
• Responses and models paths should be derived consistently from configured base path for OpenAI-compatible hosts.

Scope

• OpenAI provider endpoint selection and path mapping only.
• No UI schema change.
• No provider contract expansion.

Related

• OpenAI models lists incompatible models #5500
• OpenAI 404/400 errors #6089
• Support the OpenAI Responses API #5270
• support for openai responses api #5783