How is a vulnerability not a bug? Surely you didn’t design the software to enable exploits on your users, right?

Vulnerabilities are a special class of bugs. One that’s generally so important to fix that they get a special name and more attention.

That doesn’t make them less of a bug. It makes them more of one.

At Mozilla, everything is called a bug. It's what other systems call an "issue". So it's too late for your terminology at Mozilla. (Example: I have a bug to improve the HTML output of my static analysis tool. There is nothing incorrect or flawed about the current output.)

At Mozilla, but not everywhere: exploits are a subset of vulnerabilities are a subset of bugs.

When I worked at Mozilla, _everything_ was called a bug, whether it was a software issue, a problem in the office or some paperwork missing.

Much as GitHub calls everything an "issue" and GitLab a "work item".