After looking into it, what you have is truly an allergy to cold. Refered to as 'cold urticaria'. With cold urticaria, exposure to cold temperatures causes redness, itching, swelling and hives on your skin. People with cold urticaria should avoid exposure to cold air or water as much as possible. (Source: Mayo Clinic, http://www.mayoclinic.com/health/cold-urticaria/DS01160)
This completely agrees with everything you have said, and thus proves your cold allergy.