How about this: a "vulnerability" is a "vulnerability", but after it was identified and verified to cause problem, that's when it should be called a "bug", because it could make the software do unwanted things.
At Mozilla, but not everywhere: exploits are a subset of vulnerabilities are a subset of bugs.
Much as GitHub calls everything an "issue" and GitLab a "work item".
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.