You said:
The Wikipedia pages on blah blah blah give some idea, but these pages contain limited/unclear information, and there are rather few specific high-level resources elsewhere.
Wikipedia is great for subjects you know nothing about, but if you actually know something, all you see is incoherence, misdirection, and sometimes, when lucky, actual errors.
As to actual proof verifiers, it's just the case that each verifier has a different philosophy and different mechanism behind it (there is no one foundation that they all use). So it is difficult to answer your questions (the answer itself will be too broad).
A comment though: category theory itself is not really a foundation for proof verification. It is definitely a conceptual foundation for mathematics in that it creates analogies (and analogies of analogies) -formally- between different domains of mathematics. However there is the subdiscipline of category theory called topos theory which -does- purport to be an alternative foundation (when 'foundation' is intended to be about provability). To bounce back and forth...I don't think there is a verifier based on topos theory.
