My feeling is that the specific phrase “The Cold War” should refer to that specific milieu of tense relations between the Soviet Union and the western nations and the United States in particular, that existed from about 1948 through 1991.
It was more than just that. The term “world order” has come to mean, somewhat generically, an overall and overriding global milieu of international relations that largely define the entire trajectory of world diplomacy over an extended period of time. The phrase came into use, I think, late in the 19th century to refer to the major patterns of diplomacy among European nations at the time. (Source: The book “Diplomacy” by Henry Kissinger.) I think it was Otto Von Bismarck, or one of the major political thinkers of the day, who came up with the phrase.
After World War II, a very clear-cut global pattern of international relations evolved in which the US and USSR became, not just the “powers”, but the “superpowers”, enmeshed in non-stop diplomatic and military brinksmanship, and pervasive attempts by both sides to gain influence over other nations world-wide. And then, China got well entrenched into the mix too. I consider the specific phrase “The Cold War” to refer to that whole era, and I think of it as a bona-fide example of a “world order” of the time.
When the USSR finally collapsed in December 1991, the buzz-phrases of the day were that a world order had collapsed and a new one had begun. Myself, I agreed that a world order, specifically The Cold War had collapsed, and I agreed that it was indeed a real world order – but I didn’t agree that a new one had begun. I felt it would take another 20 years or so for a new overall world order to evolve, which would not necessarily look much like the Cold War world order.
Loosely speaking, it looks I was about right. We’ve had about 25 years now without a really well-defined international diplomatic schema. A new, but very chaotic schema involving lots of international terrorism centering around the Middle Ease has emerged in the vacuum, but it seems hard to call that a “world order”. Maybe a “world disorder”.
The newly re-emerging Russia may become part of some new world order. But I don’t expect it to be all that much like the Cold War, and no doubt once it becomes more well-defined, there will be a new name for it.