Does "V contains S" have two different meanings?

Solution 1:

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.

Solution 2:

$S$ is a set of vectors in $V$ which is not necessarily a subspace of $V$. So you can for example have $S= \{v_1,v_2,...\}$ However, when you take all linear combinations of vectors in $S$, you get the $span(S)=\{a_1v_1+a_2v_2+...|a_i \in F,v_i \in S\}$, here $F$ is the underlying field of $v$, which is a subspace of $V$.

I mean you could have $S \subset V$. Take for example $V=\mathbb{R}^3$ and $S=\{(1,0,0),(0,0,1)\}$, $Span(S)=x-z plane$.