What cools hot water more - ice or cold water?

the liquid will readily mix (homogenize) both in terms of mass and of temperature. The solid will have a surface area limitation. From a rate standpoint, the liquid will achieve thermal equilibrium much faster. In effect, the cold liquid acts like mini-volumes of one molecule thickness so you have a surface area that is pretty close to the surface area of the molecules. It doesn't start that way because there is an interface between the cold and the hot, but the interface area increases rapidly as individual molecules migrate into the bulk mass.

Those ice cubes have a fairly constant surface area (interface for energy exchange) so the energy exchange is slow. Conduction versus convection, basically.

Equal masses of ice and cold water .. then ice