> any thread unifying these things
Yes, this is the critical question. And I think the authors of this "reading group" answer that in this really smooth introduction to sequent calculus:
https://www.youtube.com/watch?v=xLRUofFSq5Y
Recommended watching!
The summary (explained in the video) is that these three areas: differential programming, probabilistic programming, and logic programming are all just programming language semantics that are more specifically "languages for calculating" and thus "calculi" languages.
These three areas are also all linked to historical and current AI research. And I think AI research is the primary motivation (another underlying thread) for this collection.