The nLab page you're looking for is called <a href="https://ncatlab.org/nlab/show/factorization+system">factorization systems</a>. Here is my favorite one, which I think answers your question in some sense. In any category with finite limits and colimits, every morphism $f : X \to Y$ has a canonical factorization

$$X \to \text{coim}(f) \to \text{im}(f) \to Y$$

where $\text{im}(f)$, the <a href="https://ncatlab.org/nlab/show/image#comparison_of_regular_images_and_coimages">regular image</a>, is the equalizer of the <a href="https://ncatlab.org/nlab/show/cokernel+pair">cokernel pair</a> of $f$ (this is the "nonabelian" version of "kernel of the cokernel") and $\text{coim}(f)$, the regular <a href="https://ncatlab.org/nlab/show/coimage">coimage</a>, is the coequalizer of the <a href="https://ncatlab.org/nlab/show/kernel+pair">kernel pair</a> of $f$ (again, the "nonabelian" version of "cokernel of the kernel"). These two constructions are categorically dual and so, among other things, the coimage-image factorization of $f$ in the opposite category is the same sequence of maps but in the opposite order. 

In $\text{Set}$, the coimage and image are both the image of a function in the usual sense, but computed in different ways, which I think match the distinction you're trying to get at. $\text{coim}(f)$ is computed by constructing the equivalence relation on $X$ defined by $x_1 \sim x_2 \Leftrightarrow f(x_1) = f(x_2)$, then quotienting $X$ by it. $\text{im}(f)$ is computed in a categorically dual way, although it looks a little strange at first: by first constructing the pushout $Y \sqcup_X Y$, then isolating the subset of $Y$ of elements which are sent to the same element by both of the canonical maps $Y \to Y \sqcup_X Y$. 

It's a nontrivial theorem that these constructions coincide in $\text{Set}$. They also coincide in any abelian category, but not in general. A very instructive example is $\text{Top}$, where $\text{coim}(f)$ is the set-theoretic image topologized as a quotient of $X$, and $\text{im}(f)$ is the set-theoretic image topologized as a subspace of $Y$.