Yes I know about vasoconstriction, it has nothing to do with the situation below:
1) When you put your hand under cold water, the water is taking heat energy from your hand and making it go down the drain.
2) Your hand is now cold compared to your body.
2) You body wants to remain in a state of endothermic homoeostasis. (Just like temperature equalization in matter.)
3) To bring your cold hand into thermal equilibrium with the rest of your body, Guess where that heat will come from?.
Yep, the rest of your body.
1) When I put my hand under the tap then only the epidermis comes into contact with the water. Heat moves from the epidermis into the water. At the same time heat moves from the rest of the hand (the bulk of it) to the epidermis. Heat is also transferred from the core of the body into the hand by convection through the vascular system (and other bits).
2) The epidermis may have changed it's temperature at this point. If the heat lost to the water is greater than the replacement then the temperature will be reduced. If the heat lost is less than the heat gained the temperature will rise.
4) Convective heat transfer to the epidermis of the hand will not occur in meaningful quantities through the vascular system when there is localised vasoconstriction due the triggering of nerves registering.
Rather than not really understanding this try it yourself. Test your capillary refill now:
a) Hold one hand above the level of your heart. Depress the nail on your index finger using the other hand for 5 secs - the nailbed will go white. Still holding that hand with the depressed nail above the level of the heart (we want blood flow here not gravity) release the nail and count the time for the nail to go from white to pink.
b) Put said finger in a freezer compartment for a few minutes. And do the same experiment.
Note how the time increases. That is because there has been local vasoconstriction. There has been reduced blood flow to the periphery due to the natural response to shut down the blood flow to areas of the skin that detect cold.
If you really think that varying areas of the body don't have independant along with central control for their bloodflow you need to work harder at school or you'll fail your exams. I hope you are at school and not at university because if that's the case you really need to hit the books. This is pretty basic stuff. GCSE Level.