After Installation PbT under Drupal 10.0.9 an Update of the database with update.php gives a Warning: (!Translation from German) The database was missing schema information for the permissions_by_term module. You should manually check the engine updates and your database to see if any updates up to and including permissions_by_term_update_9001() have been skipped. The warning is displayed only one time.
But the Status Report (/admin/reports/status) shows an error message: "Mismatched entity and/or field definitions, The following changes were detected in the entity type and field definitions. Settings The Settings entity type needs to be installed."
uninstall the module "Permission by Term", restore database and install the module "Permission by term" again.
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I solved the problem by resetting the Database to a version before the first installation of Permission by Term (PbT). Then I set up one taxonomy-list with three taxonomy terms. Now the Installation of PbT works without the mentioned problems. I looks as if for the Installation of PbT you need at least one existing taxonomy term.