[release-20.0] Fix error contain checks in vtgate package (#16672)

[vitess-bot]

Description

This is a backport of #16672

[vitess-bot]

Review Checklist

Hello reviewers! 👋 Please follow this checklist when reviewing this Pull Request.

General

• Ensure that the Pull Request has a descriptive title.
• Ensure there is a link to an issue (except for internal cleanup and flaky test fixes), new features should have an RFC that documents use cases and test cases.

Tests

• Bug fixes should have at least one unit or end-to-end test, enhancement and new features should have a sufficient number of tests.

Documentation

• Apply the release notes (needs details) label if users need to know about this change.
• New features should be documented.
• There should be some code comments as to why things are implemented the way they are.
• There should be a comment at the top of each new or modified test to explain what the test does.

New flags

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Backward compatibility

• Protobuf changes should be wire-compatible.
• Changes to _vt tables and RPCs need to be backward compatible.
• RPC changes should be compatible with vitess-operator
• If a flag is removed, then it should also be removed from vitess-operator and arewefastyet, if used there.
• vtctl command output order should be stable and awk-able.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 68.68%. Comparing base (b1eeefe) to head (f00624e).
Report is 1 commits behind head on release-20.0.

Additional details and impacted files

@@               Coverage Diff                @@
##           release-20.0   #16677      +/-   ##
================================================
+ Coverage         68.66%   68.68%   +0.02%     
================================================
  Files              1541     1541              
  Lines            197903   197903              
================================================
+ Hits             135882   135938      +56     
+ Misses            62021    61965      -56     

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