what does the set containing only the zero vector actually span?

I apologize if this sounds stupid but I am struggling to grasp the following concept. I understand that the span of the empty set is the zero vector. However, what does the set only containing the zero vector span? The zero vector as well? Also, are the following two phrases logically equivalent, "The span of the empty set is the zero vector" and "The empty set spans the zero vector". I believe they do but I don't want to make any assumptions since I'm not entirely sure myself. Thank you.

This is a good question!

In the linear algebra texts that I have seen, it is usually included in the definition of a subspace $S$ that $S$ has to contain the zero-vector. This condition is included for the one purpose of eliminating the empty set as a subspace. (Does your definition do this as well?)

But the span of a set is always a subspace, and in fact, if $L$ is some set, then the span of $L$ is the smallest subspace that contains $L$. This is why it makes sense for the span of the empty set being equal to $\{0\}$, because $\{0\}$ is the smallest subspace containing the empty set (which is not itself a subspace).

Also, if the set $L$ is itself a subspace, then the span of $L$ is equal to $L$ itself - the smallest subspace containing $L$ is $L$. In particular, $\{0\}$ is a subspace, so the span of $\{0\}$ is $\{0\}$.

Regarding your two statements, I say yes, they state the exact same thing.

In general, adding to a set $S$ of vectors a vector $v$ that is already in the span of $S$ does not change the span; in other words, in this case $S\cup\{v\}$ spans the same subspace as $S$ does.

This applies in particular to $S=\emptyset$ and $v=\vec0$; you already indicated that the span of the empty set contains $\vec0$, and so $\emptyset\cup\{\vec0\}=\{\vec0\}$ spans the same ($0$-dimensional) subspace as the empty set does.