A war requires direct conflict, not the funding of another government. Right now, many different governments all over the world favor certain sides in conflicts outside of their land and have chosen to give resources to the group that they support. France, for example, is not "at war" with Turkey right now simply because they are funding different sides of a war in Libya. People agree that America fought in the Korean War and the Vietnam war because soldiers died in both wars. However, they mostly did not die at the hands of Soviet soldiers engaged in a separate "Cold War." America and the Soviet Union were both involved in many proxy wars during that period, as most superpowers tend to do, but simply because they had policy disagreements does not mean that they were fighting.
Additionally, a war should be made up of individual battles and offensives. If the Cold War was truly a war, does that make all of the proxy wars its battles? What does that then make all of the proxy wars' own battles? A war cannot be made up of smaller wars, it must consist of deliberate plans of attack against the enemy, and the Cold War never had that.