Is this World War III?
Is Iraq or the Middle East now blossoming into WWIII? Or did the Cold War count as World War III? In that case, is this one World War IV?

Here's a straightforward, definitive answer to all of those questions. If there's any debate about whether some conflict is a world war or was a world war, then it's not a world war. When it's a full-on world war, then "all signs point to yes" as the Magic 8-Ball says. You will pretty well know when it's a world war.

Did anyone call it WWIII when the Cold War was going on? Not enough to notice. Is Iraq or Lebanon now a world war? Let's see, the Coalition in Iraq includes the US, UK, Poland, Mongolia, dwindling numbers of Italians, dwindling numbers of South Koreans... Did you know that the nation with the largest number of troops per capita sent to Iraq at one point was Mongolia? It's true.

Even if you count Israel as a client state of the US, even if Hezbollah is totally a puppet of Iran and Syria, even if you throw in five or six more Middle East nations, you don't have a world war yet. No, there has not been any third or fourth world war. If you have to ask yourself the question at all, then the answer is no.