Shapeshifter, it is time you start learning some basic wiring. This is the same pop assy. you have in your Catacomb. You need to remove yours and see what is going on. Your switch contacts are that little black blob on the left. It attaches with two small straight slot screws. Remove the two screws and let the switch hang. The coil is not installed on this unit.
To remove the coil, remove the two screws at the bottom (not shown). This will allow you to remove the coil stop. With coil stop removed, the coil is going to fall off of its mounting position so look the coil over closely to see how it is setting before you remove it; You will want to reinstall it the way it came off. The coli sleeve is going come out with the coil; Pay double attention to how the coil sleeve is installed. You have been advised. You can let the coil hang or you can secure it with some tie wraps.
There will be two wires coming from the bottom of your pop assy. You have a wire splice mess going on in there. Don't cut any wires yet, but try to partially remove at least on pop so you can see what is going on. After you have a look and if you feel comfortable, cut your wiring as close to the splice as possible and remove the pop assy. Or don't cut anything. Just pull the pop from the play field and look around. There are only two lamp wires coming from each pop so there cannot be too much to look for.
You biggest challenge/headache is the lousy screws Stern used to put this pin together with. They are a shallow head bolt. They are not magnetic. They are a pain in the ass to reinstall; You will learn to miss Phillips head screws.
The good news is that your pops do not have to be lit for you to play your pin. I don't not have my Catacomb pops lit up and I have good times playing. Lit pops have just not been a priority for me at this time but I will get there.
If you run into problems, send me an email.