I see your point. Wars were fought as a part of the larger, more static conflict all over the globe. Battles of the Cold War included Korea, Viet Nam, Angola, Nicaragua, Ethiopia, Eritrea, and various scrapes in the Mid East. A different kind of war, but a global (hence world) war. I realize that conceptulalization takes some paradigm breaking. If you read "By Any Means Necessary" you will note that plenty of Americans died in the Cold war by direct Soviet Action.
The Napoleonic Wars, 1796-1815, would certainly fit the global model. Was that World War 0, or was that the Seven Years War World War O?