Where can I find a proof of or how can I prove that these two definitions are equivalent?
Definition 1: The Lie algebra of a Lie group $G \subset GL_n$ is the tangent space at $I$.
Definition 2: The Lie algebra of a Lie group $G \subset GL_n$ is all elements $g\in G$ with the property that for all $t \in \mathbb R$ the element $e^{t g}$ is also in $G$.
Definition 2 is incorrect: In general, the Lie algebra of $G\subset GL_n$ is not a subset of $G$ (or even of $GL_n$). For example, the zero matrix is an element of the Lie algebra, but it's not an element of $GL_n$.
In place of Definition 2, you could write
Definition 2$'$: The Lie algebra of a Lie group $G \subset GL_n$ is the set of all $n\times n$ matrices $g$ with the property that for all $t \in \mathbb R$ the element $e^{t g}$ is also in $G$.
The equivalence of this and Definition 1 should be proved in most elementary books about Lie groups. You can also find it in my Introduction to Smooth Manifolds (chapters 7 and 20).