Two ways of saying span Suppose $V$ is a vector space, and $v_1,...,v_n$ spans $V$. Is that equivalent to saying that $V=\operatorname{span}\{v_1,...,v_n\}$?
Yes, the two statements are equivalent.
Here is a definition quoted from Friedberg:

A subset S of a vector space V generates (or spans) V  if span(S) = V. In this case, we also say that the vectors of S generate (or  span) V.

Friedberg, Stephen H.. Linear Algebra (p. 31). Pearson Education. Kindle Edition.

You run into issues if you try to define what it means to span in another way. For example, we commonly define a basis as follows:

A basis of a vector space V is a set $(v_1,...,v_n)$ of vectors that is independent and also spans V.

Artin, Michael. Algebra (Page 88). Pearson Education. Kindle Edition.
But now consider that $S = \{(1,0), (0,1) \}$ is an independent set and spans $\mathbb{R}^2$.
If we only require that $V \subseteq \text{Span } S$ in order for $S$ to span $V$, then we get that $S$ is an independent set that spans every subspace of $\mathbb{R}^2$, and so is a basis for every subspace of $\mathbb{R}^2$, which is not correct.
Saying that $V$ is spanned by $v_1,v_2,...,v_n$ means that every vector in $V$ can be written as a linear combination of $v_i$. Hence $V$ is a subspace of $\text{span}\{v_1,...,v_n\}$.
For example, $V=\{(k,0):k\in\mathbb R\}$ is spanned by $(1,0),(0,1)$ over the field of real numbers. But, $V\subset\text{span}\{v_i\}=\mathbb R^2$.

Note: In case you add the condition $v_1,...,v_n\in V$, then $\text{span}\{v_i\}\subseteq V$ by closure property of $V$. Then it can be concluded that $V=\text{span}\{v_i\}$.