The Hot Water Heater also is a Cold Water Heater, and maybe just a Water Heater.  But after the Hot Water Heater heats up the Cold Water,  the Hot Water Heater keeps the Hot Water Hot.  Thus, the term "Hot Water Heater".  :). You see, most of the time, if left on, the unit will be heating Hot Water (this also implies that there is a Thermostat involved).  Only when first powered up will it heat Cold Water.  

Just another point of view......

Ken Powers
SM2K 262