Another reason to call the 7y/F+I Wars the true First World War: they lost initially, and later allied with the (then young) US.
So, for now, our criteria to call a conflict a World War would be:
Fighting on at least 3 continents.
Alliances and counter-alliances that trigger widening of the conflict.
The people of Yugoslavia (or Hungary-Austria back then) fight against each other.
Germans are somehow involved and march into the general direction of Moscow.
The French surrender.
The US wins.
Thus, the 7y/F+I wars were indeed a true World War, so were the wars of 1914-18 and 1939-45.
The Vietnam Wars: no world war, although the French surrendered, and Waffen-SS fought there, Yugoslavia wasn't involved and the regular German army stayed within it's borders.
Next closest suspect: the Yugo Wars of the early 90s. Let's see: Yugoslavia was involved, German Luftwaffe flew missions over the theater, but the French didn't surrender. No world war either.
The Second Gulf War: All of the usual relevant armies played, even France surrendered properly (by forbidding USAF to fly over her territory) but fighting was limited to the Middle East. No world war.
The Cold War: lots of skirmishes and proxy wars all over the place, (East) German military involved in only one minor invasion (Czechoslovakia, 1968), complex alliances, France surrenders (multiple times). We won. But no shot was fired in Yugoslavia. No world war (but that was a narrow decision).