By definition, the Cold War was the period between 1945 and 1990 when the powerful countries did not go to war against each other but yet they tried to build up their military strength. There were wars with secondary countries and there were clashes, usually accidental, but this was a time of general peace.
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Strictly speaking there were no battles in the Cold War, that's what the term means. The two sides faced each other with ever growing arsenals of conventional weapons and weapons of mass destruction, knowing that each had the power to wipe the other off the face of the Earth, but in the knowledge that they too would be wiped out by retaliation from secure launching sites.
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Vietnam was a "hot" battle of the "Cold War". The Korea was the first "hot" battle of the "Cold War."
Cold War=military stand off (no shooting, no dying)
First Battle of Bull Run (First Manassas) : the first major battle of the American Civil War .
Vietnam War, the second "hot" battle of the Cold War.