Those were more like jostling for influence and prestige than real wars. If they were real wars Koreans and Vietnamese would fly the American flag today.
I think the moniker "World War" should apply when you have total war between all of the dominant powers on the planet.
It does. The next time there's an all-out military war across the globe, it's going to be known as "WWIII," because that's what everyone's going to call it. Just as there was no government decree that the attacks on the WTC and Pentagon were to be forever referred to as "9/11," no mere set of intellectuals is going to tell the public that we have to start calling the Cold War by a different name. Whatever name comes to the forefront of our culture is what will stick.
If there was a full nuclear exchange tomorrow morning between the U.S. and Russia, I don't think those of us that are left would be calling it "World War V."