How to close a branch by default after a pull request has been merged in Bitbucket?
What I really want is to add a comment to the accepted answer, but I don't have enough reputation to do so. So I'll note it here. I've closed BSERV-10344 because it's in the wrong tracker. This question is about Bitbucket Cloud (bitbucket.org), not Bitbucket Server (the behind-the-firewall version). The correct issue tracker for the feature request is at https://bitbucket.org/site/master/issues.

Also, re: "I'm not sure why they use different language for that", it's because Bitbucket Cloud supports Mercurial as well as Git. In Mercurial it's not possible to delete a named branch; they can only be closed. Since its support for Mercurial predates its support for Git, the terminology used is to "close" the branch.