What makes a war be classified as a "world" war?

They are decided conventionally. For example Americans called "European War" instead of WW1 at the time. There is no specific criterion for it. However, for a war to be world war it should include most powerful and populous countries, multiple continents. And they generally include multiple conflicts between those countries.

The term "World War" is a fairly modern invention -- one source says it was coined in 1940 to describe the then-ongoing wars in Europe and the Far East and retroactively applied to the Great War to name it WWI. (Note that there is no claim that the phrase "World War" was first used only in 1940. the claim is that it first came into general use about then to name WWI and WWII.)

There is no formal definition of what a "World War" is, though many have been proposed. No one is in charge of deciding what was a World War and what wasn't. There is no consensus beyond what nearly everyone agrees: that World Wars are big and that there have been at least two, both in the 20th century.

I have a book -- which, unfortunately, I can't locate -- titled something like "The Nine World Wars" which makes a fair attempt to look back in history and to develop a definition of "World War" which fits our use of the term and to examine what other conflicts fit it.

As I recall, its definition was that to be a World War, a conflict must (1) be fought over most of then then-known world, (2) must include most of the major powers throughout that known world, and (3) must be a "big" conflict which fully engages the combatants. (Personally, I find this to be about as workable a definition as we're ever likely to have.)

So, contra Wikipedia, the Punic Wars do not qualify, since, while they were certainy big wars, they were confined geographically to the western Mediterranean, and only involved two of the world's great powers. In fact, no civil war and no war between neighboring states that remains mostly confined to the territories of those two states could qualify.

Likewise, the wars of prehistory don't qualify, mostly because we know so little about them, and because their "known world" was so very small. YMMV.

What does qualify besides WWI and WWII? Perhaps the wars between Greece and Persia since they culminated in Alexander's conquest of most of the known world -- all except the barbarian West and India which he attempted to invade and failed.

The Seven Year's War certainly qualifies, as it was fought over most of the globe -- as noted by @Spencer, above, its first battle was in Pennsylvania and last was in India. It was a huge war and nearly all of the great powers were included in the festivities.

Ditto the Napoleonic Wars: While fought mostly in Europe, battles were fought over the world's oceans and on at least four continents. Only China and Japan managed to remain outside the conflict.

It's been argued that the Cold War was a World War. It certainly was big -- it cost more than any previous war -- and it certainly included all the Great Powers and it certainly was world-wide...but was it a war?

The list will inevitably be debated. To start with, there is no bright line dividing World Wars from "ordinary" wars. Secondly, the criteria are necessarily to some extent subjective, e.g., what counts as a "Great Power"? Finally, to you slice and dice or lump? (For example, a reasonable argument can be made that the Wars of the American Revolution, the Wars of the French Revolution and the Napoleonic Wars are all phases of one 40-year-long war.)

World War I was originally called Great War, only after Second World War the denomination was changed. Hence is quite probable than in the future the names will still be subject of change.

Some authors talk about European Civil War instead of World War to cover the period from 1870 to 1945 or 1914 to 1945, including here Russian and Spanish civil wars. Others say that this period was actually a religious war between communism and nationalism.

Summary: Names of events and periods are constantly changing according to new theories, sources of information or events.

A war involving many large nations in all different parts of the world. The name is commonly given to the wars of 1914–18 and 1939–45, although only the second of these was truly global.

Is the definition according to the dictionary, however there are no specific criteria involved. The reason that WW1 and WW2 are considered "World Wars" is because the main countries involved in the wars e.g. Britain, France, Germany also pitted their empires into the war.

The British Empire alone in 1920 (shortly after the end of WW1) covered roughly 1/4 of the Earths surface and population and during both WW1 and WW2 fought in theaters across multiple continents, thus making it a truly global war.