[3.10] gh-143930: Tweak the exception message and increase test coverage (GH-146476) (GH-148045) (GH-148051) (GH-148053)

(cherry picked from commit cc023511238ad93ecc8796157c6f9139a2bb2932)
(cherry picked from commit 89bfb8e5ed3c7caa241028f1a4eac5f6275a46a4)
(cherry picked from commit 3681d47a440865aead912a054d4599087b4270dd)

Co-authored-by: Łukasz Langa <lukasz@langa.pl>