OEM. Original Equipment. As opposed to an aftermarket upgrade.
    Plainly put, the pickups that came in the guitar from the factory.

    Regarding your symptoms, I don't think soldering while the battery was in the circuit would necessarily do anything bad, after all, you're just closing and opening the circuit by cutting and then resoldering anything. Additionally, if the guitar was unplugged, the circuit should have been open (non functioning) anyway. But I've never done soldering work with an active setup while the battery is connected, so I can't say anything as a matter of certain fact.

    You should probably remove the battery, unsolder your connections, redo them carefully, and then put the battery back in. It could always be one of your connections isn't perfect. But if you still do that and the problem persists, maybe you did cook one of the preamps.