Earlier quoted context omitted.
This analogy doesn't hold for "WWIII" though, right? By this analogy, I would call the current crisis "Cold War II".
World wars involve many countries, and Cold War definitely qualify. Today's war is a pretty active, quite large "hot" war, which also involves many countries - even though most fight by proxy.
I think if there is any useful distinction between "hot" and "cold" world wars then it's most likely whether super powers are in direct military conflict with each other or whether military confrontation is "only" through proxy wars.
Note that the original cold war wasn't very "cold" for much of the world either - the only thing that didn't happen was direct millitary confrontation between the US and USSR. Nevertheless there were lots of local conflicts and proxy wars where each bloc was backing a faction.