[3.14] gh-132775: Clean Up Cross-Interpreter Error Handling (gh-135492)

In this refactor we:

* move some code around
* make a couple of typedefs opaque
* decouple errors from session state
* improve tracebacks for propagated exceptions

This change helps simplify several upcoming changes.

(cherry picked from commit c7f4a80079, AKA gh-135369)

Co-authored-by: Eric Snow <ericsnowcurrently@gmail.com>