As for the "general wilderness permit", I don't think there IS such a thing. Wilderness permits are written for you to start on a specific trail, so since you are starting up the main Mt Whitney Trail, the Mt Whitney overnight permit is the permit required.
Be sure to ask about overnight permits when you show up at the Visitor Center to pick up your permit. If there are any available, you can trade. Since there is so much snow, there are lots of cancellations these days, so your chances are really good.
