You can fix the situation for the existing websites by restarting Ajenti, and then detaching unnecessary DBs

Fixed, uploading update

I'm at loss... your websites are completely separated and have different roots... how can a file then appear on both at once? Can this be a browser caching issue? Do you have any extra configuration in /etc/nginx.custom.d? Other Ajenti V websites?

А зачем index.php во флагах?

Проверьте /var/log/nginx/...error.log

You can either configure domain and subdomain at your registrar's panel, or only set up a NS entry pointing to your DNS server, and then configure everything on your own DNS server.

You did everything right, and exactly this (subdomain affecting main site) is the strangest thing. Could you please also post the generated /etc/nginx/conf.d/website.conf files with and without the subdomain website enabled?

What is the output of sudo nginx -t?
Yes, create the session dir, it should fix your main website (whose problem isn't really related to subdomain).
Also try fully restarting nginx: sudo service nginx restart.

Have you checked the Configuration checks on the main Websites page?
Also check if /var/lib/php/session exists