Add a `MessageStarted` flag to `ConvertOpenAIResponseToAnthropicParams` to ensure the `message_start` event is emitted only once during streaming.
Refactor response handling to detect streaming mode via the `stream` field instead of the `object` type, simplifying the branching logic.
Update the streaming conversion to set `MessageStarted` after sending the `message_start` event, preventing duplicate starts.
These changes improve correctness of streaming response handling for Claude integration.