A lot of the confusion between skillets and pans comes from their names. Skillets, for one, are also called frying pans or frypans. In addition, the word “skillet” is frequently used in reference to cast iron pans whether or not they are actually cast iron skillets.
No, carbon steel pans are not dishwasher safe, as they can rust in moist environments. Instead, carbon steel frying pans should be hand-washed with warm water and a soft-bristle brush or non-scouring sponge.