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Talking in terms of sets, I would take the above to mean $S in V$. But my course's notes says

Let $S$ be a subset of a vector space $V$, the span of $S$, denoted $Span(S)$ is the smallest subspace of $V$ that contains $S$.

Which confuses me, because $V$ contains vectors, whereas $S$ is a set, not a vector, so by my definition, $V$ cannot contains $S$.

So by "$V$ contains $S$" I assume it means $S subseteq V$, right? Is this considered correct also?

Yes. Unfortunately, "$x$ contains $y$" is ambiguous: it can mean either $yin x$ or $ysubseteq x$. Some writers make this distinction between "contains" and "includes": a set contains its elements and includes its subsets. Unfortunately, some do just the opposite, and say that a set includes its elements and contains its subsets. Sorry about that.

$S$ 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 $S={(1,0,0),(0,0,1)}$, $Span(S)=x-z plane$.