We know that an object is warm or hot if it has heat energy in it. And an object is cold when it does not has heat energy. So, hot of cold are two states which are defined by presence or absence of heat energy.
When we hold a cup of ice water, cold can not move from ice water to our hand. Because cold is jus absence of heat, So, our hand becomes cold because of cold moving from ice water to our hand is not correct.What actually happens is, heat energy moves from our hand to the ice water by conduction.
Heat energy always moves from hot place to cold place. As heat energy is transferred from or hand to ice water, there will be a large decrease of heat energy in our hands. We know, absence of heat energy is cold. So, our hands become cold.