chore: Replace deprecated gen_ai.responses op with gen_ai.chat

[alexander-alderman-webb]

Description

Contributes to #7950

See getsentry/sentry-conventions#703

Issues

Reminders

• Please add tests to validate your changes, and lint your code using uv run ruff.
• 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 📊

✅ 55725 passed | ⏭️ 2872 skipped | Total: 58597 | Pass Rate: 95.1% | Execution Time: 152m 14s

📊 Comparison with Base Branch

Metric	Change
Total Tests	📉 -79240
Passed Tests	📉 -74835
Failed Tests	📉 -1
Skipped Tests	📉 -4404

➖ Removed Tests (1)

View removed tests

• test_no_integration[False-False]
  • File: tests.integrations.litellm.test_litellm

All tests are passing successfully.

✅ Patch coverage is 100.00%. Project has 2063 uncovered lines.
❌ Project coverage is 90.13%. Comparing base (fe39dbb) to head (d8d670f).

Coverage diff

@@            Coverage Diff             @@
##        master       #PR       +/-##
==========================================
- Coverage    90.28%    90.13%    -0.15%
==========================================
  Files          202       193        -9
  Lines        26639     20902     -5737
  Branches      9926      7206     -2720
==========================================
+ Hits         24050     18839     -5211
- Misses        2589      2063      -526
- Partials      1509      1232      -277

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