There were several names for the war depending on the belligerent. Here are a few:
- Prussia and Austria: Dritter Schlesischer Krieg (third Sileasian war) (refers to Austria trying to reconquer Silesia)
- France: La guerre de la Conquête (War of Conquest)
- Britain: French and Indian War or Great War for the Empire
- Sweden: Pommerska kriget (Pomeranian War) (refers to Sweden trying to recover territory in Pomerania lost in 1720)
- India: Third Carnatic War (refers to the carnatic region)
- Spain: Ocupación británica de Manila (British occupation of Manila) (refers to the invasion of the Phillipines in 1762)
- Russia: Third Silesian war (according to @Michael)
As you can see, for the continental European powers this was mostly considered a regional war for a certain territory. Only for France and Britain this was a truly global conflict. Spain entered the war in 1762, shortly before its end. This is reflected in the name as well.
On the origin of the term "Seven Years War"
A quick search of Google Books reveals a reference in the Critical
Review of July 1787, which discusses Captain Tielke's history of the
Prussian, Russian and Austrian conflicts of 1756 to 1763. The article
refers to this history as "what has been styled, the seven years war".
So the term was certainly in use within 20 years of the war itself
[In Russia] it gained the name "Семиле́тняя война́" (Seven Years' War)
in the 1780s