Yep! In this case, I think you end up constructing, slightly more specifically, the isogeny whose kernel is exactly the cyclic subgroup generated by the point R

Yep! In this case, I think you end up constructing, slightly more specifically, the isogeny whose kernel is exactly the cyclic subgroup generated by the point R (i.e., phi(S) is 0 iff S is a power of R). There are explicit formulas ("Vélu's formulas") that let you compute an isogeny from its kernel. Looks like the paper goes into some depth about how to do that computation efficiently, and how to ensure that you choose a cryptographically suitable point R.