The Trigger Setup option "Do not update commit status" does not work. When I use this option, "Update commit status during build" is automatically added and therefore the status on Github is updated.

See the screenshots:
when_saved.png .. the state when I'm saving the configuration
after_reload.png .. when I leave the configuration and get back to it, there's "Update commit status during build" added
github_test_results.png .. the only workaround I found: rewrite the context to whatever (like "issue JENKINS-39449") and leave --none-- for URL. Still it clutters the list of test results.