I have solved the issue. The max_execution_time (the time limit for server requests) was too short and seized the database update after 4.7 was installed.
I have just downgraded the WordPress version back to the previous version 6.1:
– WP-includes and WP-admin folders were overwritten along with the root files of the WordPress install (robots.txt and wp-config.php are not affected at all as they are not included in the fresh WordPress release files).
After the downgrade all the categories are fine, just the news articles that were published in the meantime had “Uncategorized” category that I easily removed and assigned them correctly in the dashboard.
The solution is easy – thanks for the help!