"For every" $x\in S$ would be $\forall x\in S$ which it's same as "for all" $x\in S$. But, is "for some" is same as "there exist"? It seems Yes, but is it Yes for every time? In several texts I found both use of "for some" and "there exist", not just one of them.
As an example:
"There exists" does not mean there is only one, i.e. "There exists" is different from "There uniquely exists." So since there is no specification that there is exactly one such object (until proven otherwise), you can read the "there exists" as "for some" if you like.
It means the same thing. It's just for esthetic reasons that authors try to vary their vocabulary.
