Painting - Which primer for over joint compound?

This question is probably redundant, sorry. I put a skin coat of joint compound on my walls. In some places it got more than one coat because the walls needed to be screened.
What is the "right" primer?? I bought some Zinsser Bins 3&1, then my friend said that was the wrong primer for that. He said I should use "drywall" primer. What is the difference and which should I use??? I'll be putting Ben. Moore, eggshell for topcoat.
While I'm at it, I have to do the woodwork too. Right now it has a high gloss white paint on it. Can I paint over that with the Zinsser primer without all that sanding, or do I have to take the shine off first???

Zinsser makes a few different primers
"B-I-N" would have been the wrong primer, but would work OK
B-I-N is really a stain sealer, problem area type primer

"1-2-3" would have been the correct primer
It works great on new drywall

You should always scuff sand glossy surfaces before painting/priming