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There were several names for the war depending on the belligerent. Here are a few:
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"
From @stevebird
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
From @MichaelSeifert
[In Russia] it gained the name "Семиле́тняя война́" (Seven Years' War) in the 1780s