For me the height of the Cold War was the first Nuclear Test by the Soviet Union on the 29th of August 1949.
This was also probably the start of the 'cold-war' itself. Before the Soviet Union has tested a live Nuclear Weapon, the United States was flexing it's muscles all over the world based on it's economic and military might.
However, the Soviet Nukes proved that there was another player on the global scene with the same weapons as the United States.
Subsequently Soviet Union couldn't, and in effect did, challenge America's dominance in every corned of the world for the next 40 years.
For me, this precise moment was more important than the 'Cuban Crisis'