The United States and the Soviet Union avoided nuclear war because they did not engage in with one another. However, both chose to fight wars by providing military assistance to their allies in smaller conflicts.

indirect war,

proxy wars

Explanation:

Indirect war refers to the clash between the US and the USSR in world politics, economy and geopolitical affairs that lasted from 1945, til 1991.

Shortly after the famous meeting in the Yalta Conference, the leaders Churchill and Roosevelt showed some differences with Joseph Stalin which wanted to impose a particular vision in global politics. Since the USSR developed also nuclear arsenal by beginning of the next decade 1950, a nuclear threat arose as a means to reshape wars. However the arms race led the world in a constant tension that would result in the policy of mutual assured destruction. Since conflict between these two superpowers would result in global perish they went to engage in indirect proxy wars, supporting movements that were compatible to their interests.

The USSR backed up many uprisings and established puppet states in Eastern Europe. The same ocurred in Middle East Asia were nationalists and independentists often resolved to the military assistance to outweight the factions they opposed. Vietnam, Cuba, are the iconic instances we know and learn frequently.

The US did the same, often causing political shifts from goverments that turned into dictatorships: Nicaragua, El Salvador, Chile, just to mention a few.

In both cases proxy wars became an arena for fiercful political fights, propagana was abundant and ultimately led to armed confrontation.