Actually PADI call it "Advanced Open Water", but many people drop off the last two words. Perhaps a better word might have been "Progressive", but it's a bit late to change it now. Some agencies do have an "Advanced Diver" qualification, and that is a completely different kettle of fish. It REALLY is advanced, and is hard to get.
Yeah, I left the "Open Water" part out. But we need to get more dives regardless of training or not. This was a good side discussion though, has given us both good food for thought, as we plan the trip and future diving beyond it. We appreciate everyone's thoughts on it.