As others have said, sounds like a head gasket. The oil will be contaminated with coolant and look milky. It's common for engines to overheat when water pump fails and the overheating can warp the head or blow out the gasket. Had the same thing happen in my Corrado years ago when the plastic water pump impeller disintegrated. By the time the dummy light went off, the damage was done. Had to replace the head gasket, water pump, and aux pump, which didn't like eating little pieces of plastic and was munched up .
I also got white smoke with broken ringland in my 08 STI. It has more of a blue-ish tint to it though.
Looking at your oil should give you a good idea of where to look. If oil is low, but normal looking, that's likely oil smoke. If oil is milky looking, it's head gasket/head related.