Cuban Missile Crisis - How was nuclear war avoided?

I think that this war is avoided because the President of United States was Kennedy. If it wasn't him then the whole event might turn out the other way which has a high possibility of having a war with the Soviets since most of the generals wanted a war. Kennedy used defensive ways instead of offensive ways. Also, because the United States agreed to take back the missiles in Turkey and traded for the ones that the Soviets have in Cuba.

Still, I think the Soviets didn't want a nuclear war too. When the US said they want something before tomorrow, Soviets always give the stuff they want on time. I think a war wouldn't happen if both sides are against it.