Equivalence of Definitions of Contractible Space

Lemma
When $\bf Top$ is endowed with the cartesian monoidal structure, the terminal object $*$ (i.e. any singleton set, endowed with the discrete topology) acts as monoidal unit.

Remark
By the lemma above, there is a bijection of sets $\hom(*,X)\cong X$ between the set of functions $* \to X$ and this bijection can be promoted to a homeomorphism transporting to $\hom(*,X)$ the topology on $X$.

Note that the Lemma above and the subsequent remark hold in full generality, even when we do not restrict to a cartesian closed subcategory of $\bf Top$.

Proof of the equivalence
Assume that $X$ is contractible according to definition 1. Then every $x_0\in X$ is represented by a unique $\lceil x_0\rceil : *\to X$ such that the composition $X \overset{t_X}\to * \overset{\lceil x_0\rceil}\to X$ is the constant map $X \to X : x\mapsto x_0$. By assumption, there exists an $x_0$ such that the constant map is homotopic to the identity of $X$, which means that $t_X$ and $\lceil x_0\rceil$ are mutually inverse homotopy equivalences (the composition $*\to X\to *$ is evidently the identity of $\{x_0\}$).

Similarly, assume that $X$ is contractible according to definition 2. This means that the terminal map $t_X : X\to *$ has an homotopy inverse $p : *\to X$ such that $X\to *\to X$ is homotopic to the identity of $X$; but according to the lemma above, there is a unique $x_0\in X$ such that $p = \lceil x_0 \rceil$, so that again the composition $X \overset{t_X}\to * \overset{\lceil x_0\rceil}\to X$ is the constant map at $x_0$.