The term "wet shaving" is used to differentiate the shaving process involving a brush and a soap or cream, from shaving with shaving foam, or using an electric razor. It might as well have been called “brush shaving” because it speaks more to the point but is not wrong in assuming that more water is required.
However, the act of wetting your face occurs in shaving with foam and the act of washing off your face after a shave occurs across all three processes, so the only extra water that gets used occurs when you wet your soap with a few drops of water. The water used to wet your brush can simply be the water you store to wash off hair from your razor in between strokes. So please do not feel guilty about consuming more water if you “wet shave”, but please do feel guilty if you have a habit of using running water to shave as opposed to using your sink or a bowl to store and recycle the water you use while you shave.