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

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