ref(openai): Separate sync and async Responses patches

[alexander-alderman-webb]

Description

Remove generator pattern to allow capturing exception info on AI Client Spans in upcoming PRs.

Issues

Reminders

• Please add tests to validate your changes, and lint your code using tox -e linters.
• Add GH Issue ID & Linear ID (if applicable)
• PR title should use conventional commit style (feat:, fix:, ref:, meta:)
• For external contributors: CONTRIBUTING.md, Sentry SDK development docs, Discord community

[github-actions]

Codecov Results 📊

✅ 13 passed | Total: 13 | Pass Rate: 100% | Execution Time: 8.80s

All tests are passing successfully.

❌ Patch coverage is 0.00%. Project has 14958 uncovered lines.

Files with missing lines (1)

File	Patch %	Lines
openai.py	4.18%	⚠️ 665 Missing

---

Generated by Codecov Action

[cursor]

Cursor Bugbot has reviewed your changes and found 2 potential issues.

^{❌ Bugbot Autofix is OFF. To automatically fix reported issues with cloud agents, enable autofix in the Cursor dashboard.}

^{Reviewed by Cursor Bugbot for commit 5114533. Configure here.}