Equivalence of Definitions of Countable Set

Theorem
Let $S$ be a set.

Definition 1 implies Definition 2
Let $S$ be a countable set by Definition 1.

Then there is an injection $f: S \to \N$.

By Law of Excluded Middle, $S$ is finite or infinite.

If $S$ is finite, then it trivially satisfies Definition 2.

If $S$ is infinite, then it is countably infinite by Infinite Set of Natural Numbers is Countably Infinite, and thus satisfies Definition 2.

Definition 2 implies Definition 1
Let $S$ be a countable set by Definition 2.

Then $S$ is finite or countably infinite.

If $S$ is countably infinite, then there is a bijection $f: S \to \N$, which is injective by definition.

If $S$ is finite, then for some natural number $n$ there is a bijection $f: S \to \N_n$, where $\N_n$ is the initial segment of $\N$ determined by $n$.

Let $f': S \to \N$ be the extension of $f$ to $\N$.

Then $f'$ is an injection.

Thus in either case $S$ is countable by Definition 1.