In "Counting Convex Polygons in Planar Point Sets" (link) Mitchell et al. show an algorithm which can do this in time $O(cn^3)$.
Note however that if you need to report those polygons then you can't do better than $O(n^c)$ as there might be $\Omega(n^c)$ such polygons to begin with (consider for example $n$ points in convex position).
There is an infrequently held conference series called History Of Programming Languages (HOPL). It was held in 1979, 1993, and 2007, the fourth installment is scheduled for middle of June, 2020.
The Proceedings for HOPL-I and HOPL-II were also published as books, for HOPL-III, both the papers and video recordings of the presentations are available. (...
There is no set terminology for this. Check the definition/use of the term where it appears.
Personally, I'd call it "computation", meaning the sequence of configurations the automaton goes through. And try to be consistent with this usage.