The Stacks project

Suggested slogan: Closed immersions correspond to quasi-coherent sheaves of ideals of finite type.

The additional equivalence with “$i_*\mathcal{O}_Z$ is a finitely presented $\mathcal{O}_X$-module” follows easily from the statement (and maybe it is worth adding it?). On the one hand, if $i_*\mathcal{O}_Z$ is of finite presentation, it follows that $\mathcal{I}$ is of finite type by 17.11.3. For the converse, consider the exact sequence $$\mathcal{I}\to\mathcal{O}_X\to i_*\mathcal{O}_Z\to 0.$$ If $\mathcal{I}$ is of finite type, then given $x\in X$, we can find a surjection $\mathcal{O}_U^{\oplus r}\to\mathcal{I}|_U$ in some open neighborhood $U\subset X$ of $x$, and, hence, we can replace $\mathcal{I}$ by $\mathcal{O}_U^{\oplus r}$ in the last sequence (after restricting) while maintaining exactness. Thus, $i_*\mathcal{O}_Z$ is of finite presentation.

I agree with what you wrote, but I am going to leave this as is for now.