feat: separate OpenAI Chat Completions and Responses API handlers

- Created base OpenAIBaseHandler with shared logic (auth, retry, metrics)
- Split into OpenAIChatCompletionsHandler for Chat API
- Split into OpenAIResponsesHandler for Responses API
- OpenAiHandler now routes based on URL pattern and model type
- Maintains backward compatibility with existing tests
- Clear separation improves maintainability as APIs evolve

Addresses #8246

Author: roomote