I recently heard a Serbian historian refer to this war as "Great Vienna war from 1683 to 1699", the context was that he was speaking about the consequences this had on the Jewish community of the Vojvodina. However in German historiography this war is still referred to as "Great Turkish war" (Großer Türkenkrieg). I also saw it referred to as "Austro-Turkish war" on a website that spoke about the same topic the historian was talking about.
I wonder now how the best way would be to refer to this war. After all "Turkey" was the Ottoman empire back than and it was fighting the "Holy Liga" which was much bigger than Austria. And "Austro-Turkish war" obscures a bit that it was the Ottomans who were trying to conquer more land and gives it a weird vibe referring to two nations that actually didn't exist back than. Now it makes me think to refer to it as "Great Vienna war from 1683 to 1699" is not so bad as people know the second battle of Vienna and it circumvents this question. What do historians that know about the time period say? How should we refer to this war best for our history video project? What would be most correct? Mind you, we would prefer it to be also short.
"Great Turkish War" is, I believe, still a commonly used term. The major alternative would be "War/s of the Holy League [of 1683-1699]," but there you'd want to include the date range to disambiguate it from the mid-sixteenth-century War of the Holy League, and that makes the name a bit clunky for common use. (Which is preferable depends on your lens: "Great Turkish War" is a bit Eurocentric, but that's acceptable if the main focus is on European actors and consequences; I think "War of the Holy League" is better if you're looking for a more general or Ottoman-centered theme.)
Otherwise: "Great Vienna War" pops up occasionally in English-language historiography, but usually it's in the work of a Serbian or Croatian historian. I think that's just the term used over there, and while it wouldn't be unforgivable to use it in English, it's certainly not preferred.
"Austro-Turkish War," similarly, could work but wouldn't be my preferred term—in part because, as you said, the Austrians weren't really the only belligerents on their side, but more importantly, because there are already too many Austro-Turkish Wars and who needs another one?
Another alternative which is used in a relatively recent (2019) book on the Treaty of Carlowitz is "The Second Long Turkish War"—sometimes, just "The Long War." The risk here is of confusion with the first Long Turkish War, which, unlike this one, is pretty much always referred to by that name.