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.