What commands are executed by the Merge button?

I just wonder, because I merged it maunal local on my machine via simple command, and the branch on GitLab is still not marked as “merged” (no badge).

What commands are executed by the Merge button?

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I’ve had no problem with the Merge button executing a Merge, and then deleting the source branch ^^in Gitlab^^ if selected. You can confirm this by looking at the repo->branch page ^^in Gitlab^^, and the source branch will not be there.
Where you may be getting confused is, if you are looking at your local, it will still be there since it was only deleted in the Gitlab remote, not your local. This is expected. What is bad is that if you are looking at something like the TortoiseGit repo graph, the Gitlab remote version will appear to be still there, but it is not. To clean up your local, you need to delete the local manually, and also (re-)delete the branch at the origin from the local.
What I need help with is, what is the value of CI_PIPELINE_SOURCE when the button is clicked, or what is a way of launching the pipeline when it is clicked if the pipeline is otherwise suppressed in the .yml file.