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