docs(api): sync public integration contract to test

[alick-liming]

Summary

• Port the public integration contract from docs(api): align public integration contracts with datasource #556 #1028 (2922d47) onto latest test (12709338).
• The test baseline did not include /integration/*; add only the 9 management endpoints and their 13 missing referenced schemas to the existing bilingual aggregate/On-call specs.
• Include stored settings, decrypted integration keys, permission metadata, optional plugin-test results, and the corresponding request fields.
• Add bilingual navigation/catalog entries and the 9 integration redirects; adjust only the affected endpoint totals.

Validation

• python3 scripts/lint_openapi.py: 12 specs, no violations.
• node /private/tmp/sync-integration-docs-test-20261010.cjs: all 9 endpoints and referenced models exactly match docs(api): align public integration contracts with datasource #556 #1028; unrelated test JSON text, models, paths, navigation and catalog content are unchanged.
• node /private/tmp/check-integration-test-examples-20261010.cjs: 72 request/response examples and error response references passed.
• git diff --check: passed.
• Independent simplify and correctness reviews.

Scope / Notes

• Based on latest test, not a merge of main history. One focused commit.
• PR targets feat/public-integration-contract-test-20261010, created from the same latest test baseline, per repository branch workflow. After review, integrate this feature into test; do not merge it into main.
• SDK skipped as requested: go-flashduty has no remote test branch.
• Local Mintlify preview not run: the Mintlify CLI is not installed.