But I mean, even if the Bayesian definition covers more than the frequentist definition (which it definitely does), why not just use both definitions and understand that one application is a subset of the other application?

You'll have to ask to a frequentist :)
Bayesian use both definition (even though they call long-run frequency... well, long-run frequency), but frequentist refuse to acknowledge bayesian probability definition and methods.

but the second chapter gave me brainhurt, so I put it down for a while. I think it might be that I never took calculus in school? (something I now regret, oddly enough for the general population) So I'm trying to becoming stronger before I go back to it. Do you think that getting acquainted with Cox's Theorem in general would make Jayne's particular presentation of it easier to digest?

I skipped the whole derivation too, it was not interesting. What is important is at the end of the chapter, that is that developing Cox requirements brings to the product and the negation rules, and that's all you need.