Does "V contains S" have two different meanings?

Yes. Unfortunately, "$x$ contains $y$" is ambiguous: it can mean either $y\in x$ or $y\subseteq x$. Some authors make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some authors do just the opposite; e.g., quoting from p. 33 of C. St. J. A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965), 33–37:

To avoid confusion, we shall say that a set includes its elements and contains its subsets.