I'm going to take a risk here - I want to try to illustrate why I think the titles of 'First and Second Wizarding Wars' are suitable, even if the reasons can be a bit contradictory (that's not my fault is it?!)
'First and Second' vs 'Global'
As you alluded to in your question, history often names wars in ways that aren't necessarily numerical, but still make sense. (We all know what World War Three suggests; a future end of the world disaster is the usual thought, not a real-life war which actually predates what we call World War One but is still technically a world war).
So, Voldemort's two attacks of British wizarding society are referred to as The First and Second Wizarding Wars, perhaps asking the user to mentally continue 'in Britain' or 'against Voldemort.' Maybe even 'in Britain in modern times.'
There is a Wiki page for Gellert Grindelwald's dangerous actions in Europe (often called his 'revolution' in other aspects of the Wiki pages) titled the 'Global Wizarding War.' While it seems odd, perhaps wizardkind casually refer to 'The Global War,' 'The First War' and 'The Second War,' all knowing exactly what they mean even if the titles aren't perfect (just like our WWI, WWII, etc. The names are just fine and globally accepted, but can be picked apart quite rightly if you have a mind to).
Ways in which Grindelwald's actions may not have constituted a war
I know, I just argued that 'Global Wizarding War' is a good title, but now I'm going to talk about how it might not have been an actual war. The Wiki does the same, I'm not sure if JKR is more clear (I don't think she actually calls this a war either).
I'm just trying to reflect my findings with the key conclusion that, if nothing else, the names used for the Voldemort-related wars are suitable.
Despite there being a Wiki for Grindelwald's 'Global Wizarding War,' the Wiki for Gellert Grindelwald says: (Emphasis mine).
"The extent of Grindelwald's success in his revolutionary endeavours is unclear. He succeeded in creating a fortress, Nurmengard, to serve as his power base, apparently sufficiently impregnible that he was later imprisoned in it, inaccessible to his former supporters and breached only by Voldemort. But there is no evidence he succeeded in toppling any wizarding governments or made much of a dent in the Statute of Secrecy, as it remained stubbornly intact (though Muggle World War II was certainly a sufficiently chaotic and violent time to hide some serious breaches). In what few accounts of Grindelwald's revolution exist, his great power as a wizard is mentioned often, but there is less mention of his crimes. One murder is noted, that of Viktor Krum's grandfather, and presumably there were others, but this lack of details protects Grindelwald's moral ambiguity."
So, he may have had less impact than Voldemort and had, as I suggested in a comment, more of a reign of terror and less a war.
He had a fortress, but does that prove much in and of itself? He was building an army, but did he use it?
In Deathly Hallows, Dumbledore explains,
"I delayed meeting him until finally, it would have been too shameful to resist any longer. People were dying and he seemed unstoppable, and I had to do what I could."
So, people were dying, but does that suggest thousands in battles necessarily? He seemed unstoppable, but could that perhaps suggest that no other wizard bested him one-on-one?
The whole concept of Grindelwald's revolution feels more personal, and, unlike any war I can think of, it was conclusively ended in a one-on-one fight. Dumbledore was called upon, they battled, and Grindelwald lost. There is no suggestion that Dumbledore had to take down an army to reach him. This ended the war/revolution. This aspect does not sound war-like to me, and the name revolution is more frequently used in the Grindelwald article.
I found this a very interesting question and have wondered about it before. I did my best to find out more, found the answers inconclusive and almost left it there. In the end, I decided that this information leads to the idea that the names of the wars work, inspired by the comments that our historical titles are not always 100% accurate.
So, to answer your question: Was 'The First Wizarding War' really the first war? Almost certainly not, but the name works.