After several mishaps, I got the pickup wired to the guitar again. This time, the final one was that the live wire’s casing was frayed at the point it was connected to the pickup, and in some positions, it touched the bare ground wire next to it.
I tried to insert tiny bits of electrical tape between the two with tweezers, but it’s too tiny a space for me successfully work with. I ended up taping the wires to the outside of the humbucker in a way that they usually don’t touch each other. I’m afraid they will when something gets jostled, though. I thought about taking the pickup apart to further separate the two wires, but then I remembered this pickup cost $10. If I get one that’s maybe $40, I probably won’t have to deal with this at all.
Though, I bet they do sell expensive pickups that will eventually have this problem because it stems from it being a “single wire” pickup. Where the “single wire” is the live and ground wrapped into one casing. It looks convenient, but is actually a leaky abstraction. You’re better off with separate live and ground wires with space between them so they won’t touch when their casing fray.