Equivalence of Definitions of Well-Ordering

Definition 1 implies Definition 2
By hypothesis, every subset of $S$ has a smallest element.

By Smallest Element is Minimal it follows that every subset of $S$ has a minimal element.

Thus it follows that $\preceq$ is a well-ordering on $S$ by definition 2.

Definition 2 implies Definition 1
Let $\preceq$ be a well-ordering on $S$ by definition 2.

That is:
 * $\preceq$ is a well-founded total ordering.

By definition of well-founded, every $T \subseteq S$ has a minimal element.

By Minimal Element in Toset is Unique and Smallest, every $T \subseteq S$ has a smallest element.

The result follows.