Why is "pure" water not really pure? What about distilled water?

First, the word pure has no single meaning and each person will give you a different answer. It is like the words safe and comfortable. The definition of absolutely nothing but water molecules is a physical impossibility so we don't have to talk about that. When people say pure, they mean below a certain level of contaminants. This could mean many things depending on the application for the water. Pure drinking water is different than general pure lab water which is different than pure water for a fish which is different than pure water for medicines. See the links for info on distilled and deionized water.