I agree, I suffer with the same symptoms. If my skin gets cold i.e.. come out of a lukewarm shower in to a cold changing room or if I wear a t-shirt on a cool day I itch like mad. It is Aquagenic Pruritus. I have seen Drs (they have not heard of it) and Dermatologists (nor have they) and I have tried everything. What I have learned:
Moisturisers dont really work, it is not a dry skin issue.
Phototherapy & sun help it but not work the skin damage.
It is not a reaction to chemicals or detergents etc. There is no rash, it is not an allergic response.
It always comes on in fall (October) and goes away in summer. Never affects me on tropical holidays.
Water doesn't cause it - COLD or WARM water will trigger it though. Cold is the enemy.
I take long hot baths or showers (less effective) to get my skin very warm then dry off quickly and get dressed right away. Never take a cool or warm shower.
Wear long sleeve cotton clothes around the house in winter.
Flannel cotton PJs in winter if it is bad.
As soon as you feel itchy cover up and it will pass. I don't tend to ever feel cold but I notice when the itch starts my skin is very cool to the touch. Seems like a circulation issue triggering overactive nervous response. The fact that it is only dorsal (tops of arms and legs, back) makes me think it may be related to sun damage but that is a guess.
If you are really itching like mad try a long hot bath. Solves it every time and seems to cure it for a day or so. In extreme cases I will use Skin So Soft in the water. The oil helps to trap heat in your skin and reduce evaporation.
It sucks but it is manageable. At least there is no rash.