Generally speaking, swing is jazz, but there are some artists that are generally labelled as "swing" that compromised the jazz elements to such a degree that it's questionable that they should be considered jazz. It may be best to consider them jazz-influenced pop. For this list, I judged Glenn Miller to be just barely on the jazz side of the jazz/pop divide, and Frank Sinatra borderline at best, but both cases are debatable.
"Big band" often refers to a way of playing swing (swing can also be played by small groups), so what I said above about swing also applies to big band. (It doesn't matter in this case, but there's also some big band jazz that isn't swing.)
Sinatra sang standards, but they really were pop standards, though pop standards were often played by jazz artists too.
What is your statement about "vocal pop" in reference to? Because I'm not sure I'd call "vocal pop" its own genre either.
I always thought traditional pop and jazz were like almost the same thing? Frank Sinatra is labeled as doing both traditional pop and jazz, right? I thought traditional pop was like a sub-genre of jazz. Sorry, but I've always been a bit confused when it comes to traditional pop and jazz.