refactor: propagate `context.Context` through `Vendor` interface methods

- Add `context.Context` parameter to `ListModels` interface method
- Add `context.Context` parameter to `SendStream` interface method
- Thread caller context through `Chatter.Send` instead of using `context.Background()`
- Update all vendor implementations to accept context parameter
- Introduce `publicError` type wrapping Codex provider errors
- Normalize Codex error messages to lowercase for consistency
- Add context-aware `sendStreamUpdate` helper in Codex client
- Remove stale `context.Background()` calls from Anthropic and Azure AI Gateway
- Update all vendor test mocks and stubs with new signatures
- Pass HTTP request context from server handler into `chatter.Send`