Fix MySQL CI jobs

What does this MR do?

EE port can be found at https://gitlab.com/gitlab-org/gitlab-ee/merge_requests/13149.

1. This brings back https://gitlab.com/gitlab-org/gitlab-ce/merge_requests/27514 which broke MySQL jobs on master.
2. This also adds MySQL jobs that can be started with a specific branch name that contains mysql or commit message ([run mysql]).

Does this MR meet the acceptance criteria?

Conformity

• [-] Changelog entry
• Documentation created/updated or follow-up review issue created
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content

Performance and testing

• Review and add/update tests for this feature/bug. Consider all test levels. See the Test Planning Process.
• Tested in all supported browsers

Closes #62151 (closed), https://gitlab.com/gitlab-org/gitlab-ce/issues/62156.