During the 2000's there wasn't really an official name for the conflicts in Iraq ane Afghanistan-- people just called it the war in Iraq or the war in Afghanistan. In recent years it has come to ge more or less accepted to call them the Iraq War or Afghanistan War.
For wars in the past like World War 1 and 2, did people call them WW1/2 when they were being fought or did those terms come later? If they came later, what were they called at the time they were happening?
Also, how do wars tend be come by their proper noun name in the first place?
/u/lord_mayor_of_reddit and /u/hannahstohelit have previously written about how World War 2 got its name in English as linked by /u/jbdyer
Answers about the nature of a "World War" are in the comment below.