The world wars were called so because, simply put, they involved fighting all over the world. This was much easier to each in WWI because of massive colonial empires in large alliances. In the second world war, the colonialism aspect was still in effect, but several smaller wars were joined into one when alliances became more solid.
Obviously, two countries fighting (and there are pretty much always two countries fighting) is not a world war. NATO countries providing weapons to a another country fighting a war is business as usual for the past 70 years. This is why it was called a "cold war", as it's a series of proxy conflicts by larger powers basically playing war games with other people's countries instead of their own. E.g. The Chinese civil war, the Korean civil war, the Vietnam civil war, the Syrian civil war, etc. Large powers arm one side of a conflict to test out new weapons and tactics without committing their own soldiers (which some exceptions, like Vietnam).
Nukes are not necessary for the world to be at war (nukes were sparsely used in war so it's a rather poor indicator statistically). The usual harbinger is a sign of a broken trade network. This can be seen after the fact in both world wars. A second indicator is, obviously, most countries in the world are fully mobilized (in a state of war). Right now, Russia isn't even fully mobilized which means that they don't seem to be treating the invasion of Ukraine very seriously, which is evident based on weapons used, various tactics, and their willingness to retreat from positions. A final indicator would be that countries enter a state of total war. Total war is basically the transition from fighting armies to razing cities. All three of these indicators were present for both of the previous world wars so they are likely good indicators for a future one.
I'd also reiterate the first point: a war across Europe is not a world war. Since time immemorial, there have always been wars stretching across the continent, from the Roman Republic, to Genghis Khan, to the Thirty Years War, to Napoléon. Each of these conflicts were ultimately, localized to a single continent with a handful of solvent and sovereign countries, not most of the world's countries on each continent.
