[3.11] gh-115233: Fix an example in the Logging Cookbook (GH-115325) (GH-115355) (GH-115357)

Also add more tests for LoggerAdapter.

Also support stacklevel in LoggerAdapter._log().
(cherry picked from commit 225856ef3e6c5e4f234ede1dd118b57f6e8f6d0e)
(cherry picked from commit 91822018eeba12a6c9eabbc748363b2fd4291b30)
Co-authored-by: Serhiy Storchaka <storchaka@gmail.com>