Is all water created equally?
As regards the extreme case where the water is frozen, there is what is known as “heat of crystallization”, an amount of energy associated with the transition from solid to liquid, which doesn’t manifest as a temperature change. Depending on the substance involved, that transition may yield energy or consume it. In the case of water, energy is consumed by the melting process. What that means is that ice has to absorb some heat from its environment in order to melt, without the absorbed energy manifesting as a significant temperature change. Ice at 0 degrees C is a lower energy state than water at 0 degrees C.
If you look at things from the point of view of entropy, or losing heat, then the hot water loses its heat to the air, and the air loses its heat to the cold water. This doesn’t seem symmetrical to me. Suppose the hot water loses its heat to the air more quickly and efficiently than the air loses its heat to the cold water, and it raises the room temperature ever so slightly above the original median in the process. In this case, until final equilibrium is reached, the room temperature is closer to the temperature of the hot water than the cold water. So the hot water reaches equilibrium with the air first (at slightly warmer than the original median), then both cool (with tiny fluctuating versions of the original process) to meet the rising temperature of the cold water. Then you have final equilibrium, which, by definition, is simultaneous.
They told you wrong, GEM. Hot water has to become cold water before it freezes, with all the attendant properties of cold water, including increased density. It can’t go from hot water to ice without becoming cold water in the process (barring any very unusual conditions such as very strong electric fields).
