Why are the two definitions of a generating set of a group equivalent? $\left( G, \circ \right)$ is a group:
Definition 1: $\left<S\right>$ is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses. 
Definition 2: $\left<S\right>$ is the smallest subgroup of $G$ such that $S \subseteq \left<S\right>$.
Why are the two definitions equivalent?
 A: Let's fix some notation.


*

*Let $H_1$ be the subgroup consisting of combinations of elements of $S$ and their inverses.

*Let $H_2$ be the "smallest" subgroup containing $S$.


When we say "small" in this context, what we mean is that $A$ is "smaller" than $B$ if $A\subseteq B$. A more rigorous formulation of the second definition is something like


*If $H$ is a subgroup and $S\subseteq H$ then $H_2\subseteq H$. (i.e., $H_2$ is smaller than every subgroup that contains $S$.)


This is the definition I'll work with instead of the second one.
I want to convince you now that $H_1=H_2$. The usual way to do this sort of thing is to show that $H_1\subseteq H_2$ and $H_2\subseteq H_1$.
It should be clear that $H_2\subseteq H_1$, since $H_1$ contains $S$ and $H_2$ is contained in any subgroup that contains $S$, by definition. Conversely, any subgroup of $G$ that contains $S$ must also contain all combinations of elements of $S$ and their inverses. This is because a subgroup has to be closed under multiplication and taking inverses. In other words, any subgroup of $G$ that contains $S$ must contain $H_1$, and in particular $H_1\subseteq H_2$.
A: Assume that a group $G$ and a subset $S\subset G$ are given.
Then the set $S$ is a generating set, if $\ldots$
Definition 1: $\quad\ldots$ each element $g\in G$ can be written as a finite product of elements from $S\cup S^{-1}$;
Definition 2: $\quad\ldots$ any subgroup $H\subset G$ containing the set $S$ is in fact $=G$.
Your definition 1 is correct with the proviso that $\langle S\rangle$ should be replaced by $S$. By $\langle S\rangle$ one commonly denotes the set "generated" by $S$, i.e. the set of products described in the text of definition 1.
Your definition 2 is nonsense: Since  $S\subset\langle S\rangle$ is always true it just says that $\langle S\rangle$ is the smallest subgroup of $G$ – exactly the contrary of the intention.
