It's referred to as a war, and it certainly spanned the globe. There were even "hot flashes" like Korea, Vietnam, and Afghanistan.
What's stopping the decades long conflict from being referred to as World War III?
The main thing preventing the Cold War from being termed "World War III" is the fact that throughout the nearly fifty year period of heightened political and military tension, the two key "belligerents" of this ideological conflict, the United States and the Soviet Union, never engaged in open war with one another. While there were armed conflicts throughout this period they never saw direct engagement between either country: while American forces participated openly in the Korean War, Soviet involvement was largely limited to ideological and material aid for North Korea, with some limited military support in the air war; the situation in Vietnam was fairly similar as it saw American forces engaged in a protracted ground war, with the Soviet Union once again providing ideological and material support for the government in Hanoi and the NVA; conversely, the situation in Afghanistan devolved from a Soviet invasion into a drawn out occupation and open conflict with a number of groups made up of ethnic fighters, including the Mujaheddin, with the US eventually providing material support to the Afghan fighters.
As I have outlined above, neither side took a major role in challenging the other during these "hot flashes" you mentioned. Additionally, a number of the conflicts which erupted during the Cold War can be termed as "proxy wars," (with the slew of Arab-Israeli wars taking place from the 1950's-1970's serving as a clear example) which saw armed conflict between the client states of both the US and the Soviet Union and ranged from the Middle East, to Africa, and Southeast Asia. Therefore it would be best to refer to this period as strictly the Cold War, as the conflicts which took place during it do not provide the proper political or military context to refer to as "World War III."
Source:
Walter Lafeber, America, Russia and the Cold War 1945-2006 (Up to date and revised version, which provides a good historical background to the beginnings of the Cold War, as well as information on the number of armed conflicts which took place during it.)