I don't believe there is any set standard- rather, it's determined more by popular usage in the aftermath and the angle which people in a given country wish to put on it. For this reason, many wars have various names.

For example, we today would generally refer to the American Civil War as just that. However, there are other names in circulation present a different viewpoint. Calling it the 'War Between the States' casts it more as a conflict between two polities (thus validating the CSA's claims to nationhood) than as an internal conflict within the U.S. Taking things even further, the name 'War of Northern Aggression' presents a very strong pro-Southern viewpoint, part of the veneration of the 'Lost Cause' that persists today. Clearly, opposing sides of a war may use very different terms to refer to it.

What about third-party names? The Second Sino-Japanese War (the WWII-era Japanese invasion of China) is known as such in English, but what about Japanese and Chinese? During the war, the Japanese called it the China Incident, a term which is sometimes still used today, but is of course loaded with tension, especially in light of the still-fraught relationship between Japan and China. In China, it's called the War of Resistance against Japan- very clear-cut. What's interesting is that, in the West, it's called the 'second' war, thus linking it with the first Sino-Japanese war, a connection that's really isn't made by either Japan or China. One reason for this is that the first war was fought against the Qing Empire, not the Republic of China, a detail that gets somewhat glossed over in the 'big picture' Western view. Another interesting facet is that this is generally viewed as being part of WWII in Western eyes, while the belligerents (China especially, Japan somewhat less so) see it more as an independent event, albeit one tied up in global geopolitics of the time.

For a final example, here's one of my favorite examples of war nomenclature. The Opium Wars were fought between Britain and China primarily over trade rights (the Chinese had banned the opium trade and had long prevented free foreign access to Chinese markets). But why are they called the Opium Wars? They could have been the Wars to Open China or the First and Second Sino-British Wars, or what-have-you, but they weren't. It's also critically important that this is what they're called in English, because that's the name the British press and people gave to them, despite efforts of the government to the contrary. This illustrates the great conflict within the UK over a war that was basically fought so that their country could keep a massive drug business going.

War naming conventions are really far from a cut-and-dried thing. They can vary widely depending on who you ask, especially if you talk to different belligerents of the war. As we've seen, they often become another tool of national policy, one that can be used by the government to express their position and sway the public, or co-opted by dissidents to express their dissatisfaction with official actions. They can change as time goes on (remember the 'war to end all wars'?) as people and historians reevaluate what occurred, and they continue to shape our perceptions of world events- if you don't believe me, ask yourself what the 'Global War on Terror' really means.

To the US, our 'Vietnam War' is generally known as the 2nd Indo-China War (with variations). The 1st one being the "French' war. All depends on who is writing the history.