The standard definition of a power object seems to be: objects $\mathcal{P}X, K \in \mathbf{C}$ and a monic $\in: K \hookrightarrow X \times \mathcal{P}X$ such that for every monic $r: A \hookrightarrow X \times B$, there is a unique morphism $\chi_r: B \to \mathcal{P}X$ such that $r$ is the pullback of $\in$ along $\mathrm{id}_X \times \chi_r$ (e.g. here).
I'm struggling to see how this faithfully captures the notion of a power set. Intuitively, this idea should be captured by a natural isomorphism between subobjects/monics $r: A \hookrightarrow X$ and elements $1 \to \mathcal{P}X$. Which would lead to a definition such as the following:
Alternative definition: a power object of $X \in \mathbf{C}$ is objects $\mathcal{P}X, K \in \mathbf{C}$ and a monic $\in: K \hookrightarrow X \times \mathcal{P}X$ such that for every monic $r: A \hookrightarrow X$, there is a map $\bar{r}: 1 \to \mathcal{P}X$ such that $r \times \bar{r}: A \to X \times \mathcal{P}X$ factors through $\in$, and is universal for this property.
Intuitively, if $r$ represents the inclusion of a subset $A \subseteq X$, then $\bar{r}$ is the inclusion of $\{ A \}$ into $\mathcal{P}X$. $r \times \bar{r}$ factors through $\in$ since all elements of $A$ are elements of $A$, and $A$ is the universal set with this property.
Is this an equivalent definition? Is it equivalent under some assumptions, such as existence of finite limits/colimits?