In Russia, for instance, there are 11 grades of school education: you get one diploma after 9 grades and another diploma after 11 grades. Is it all "middle school"? Or is it "middle school" till grade nine (which gives you a middle school diploma), and "high school" afterwards (which gives you a high school diploma)? What is "college" in international settings? In Russia, we have the word "колледж", but it refers to a fancier type of school rather than a higher education facility. "Institute", how is that word supposed to be used? The same way "институт" is used?
There isn't any correct way to refer to all the different levels of all the school systems in the world in English.
I imagine the English-speaking locals in each country develop their own jargon, or more likely just use the local word.
Where I'm from, the term "middle school" has no meaning because we go from elementary school straight to high school.
The word "college" in the US means something different from the rest of the English speaking world, so there's definitely no standard for what it means in the non-English speaking world.
So if you're trying to describe the first diploma that students receive after 9 years of studying, for instance, you should just describe it in those words, or if the precise meaning isn't important enough to describe, then just pick a words that's close enough to give the approximate age of the students who attend.
According to Wikipedia "Middle school in Russia covers grades 5 to 9". In countries where the term is used, the exact age range varies.
In British English college usually means an institution of further education, though it is sometimes part of the name of a secondary school. The separate institutions within the universities of Oxford, Cambridge and Durham are also called colleges.