Is there a name for a section and a retraction together? I call $(s, r)$ such that $r\circ s =\operatorname{id}$ a split morphism. (Other names are “section-retraction pair”, “split pair.”) Notice that this is not a section, because any section is one morphism. I want to make sure that no official name exists thus I can invent it myself. Do you know the name?
The same question about isomorphisms. “Isomorphism pair”, “isomorphic pair”?
The distinction may be quite apparent. Every split morphism is a categorical diagram whereas a section is not.
 A: Very late answer...
According to Borceux-Bourn [*] this is a "point" (if you assume categories are named after their objects in some sense).
The points in a category $\mathcal A$ together with pairs of morphisms making the obvious diagrams commute form a category $\mathsf{Pt}(\mathcal A)$, the category of points of $\mathcal A$. Assuming $\mathcal A$ has pullbacks along split epis, the "codomain functor" $\mathsf{Pt}(\mathcal A)\to \mathcal A$ is called the fibration of points (it is a fibration in this case). The fibration of points is useful to classify various notions between (let's say) $\mathcal A$ being Mal'cev and $\mathcal A$ being essentially affine (Abelian minus [pointed & has cokernels]).
I suppose the name "point" comes from the fact that the fiber $\mathsf{Pt}_A$ of the codomain functor over $A$ is (essentially) the category of pointed objects of the slice category $\mathcal A \downarrow A$. The category of pointed objects of a category $\mathcal B$ with a terminal object $\mathbf{1}$ is the coslice category $\mathsf{1} \downarrow \mathcal B$, which is, you guessed it, pointed.
[*] F. Borceux, D. Bourn - "Mal’cev, protomodular, homological and semi-abelian categories"
