I finally fixed the issue.
From the beginning, the real problem was at the socket permission.
I put down bellow my walkthrough, for everyone who need to understand it in the future.
1°) Check permissions on socket file (Apache has no right as user for the moment)
root@server:/run/php# ls -l
…
srw-rw---- 1 dev www-data 0 Feb 4 15:38 php8.1-fpm-websitename.sock
2°) Have to give permission to Apache’s user (aka www-data)
root@server:/run/php# chown www-data:www-data php8.1-fpm-websitename.sock
root@server:/run/php# ls -l
…
srw-rw---- 1 www-data www-data 0 Feb 4 15:38 php8.1-fpm-websitename.sock
3°) Check PHP 8.1 FPM status (failed)
root@server:/run/php# service php8.1-fpm status
…
Active: failed (Result: exit-code)
…
… WARNING: Nothing matches the include pattern '/etc/php/8.1/fpm/pool.d/*.>
… ERROR: No pool defined. at least one pool section must be specified in c>
4°) Trying to restart Apache (failed)
root@server:/run/php# systemctl restart apache2
…
Active: failed (Result: exit-code)
…
… WARNING: Nothing matches the include pattern '/etc/php/8.1/fpm/pool.d/*.>
… ERROR: No pool defined. at least one pool section must be specified in c>
5°) Navigating to PHP 8.1 Pool.D (backup of conf files I made), and restauring the conf file (said in the error log “Connection refused : AH02454”)
root@server:/# cd /etc/php/8.1/fpm/pool.d/BACKUP_conf/
root@server:/etc/php/8.1/fpm/pool.d/BACKUP_conf# cp BKP__websitename--cnf /etc/php/8.1/fpm/pool.d/websitename.conf
6°) Checking if conf file is in right place
root@server:/etc/php/8.1/fpm/pool.d/BACKUP_conf# cd ..
root@server:/etc/php/8.1/fpm/pool.d# ls -l
…
-rw-r–r-- 1 root root 1243 Feb 5 17:42 websitename.conf
7°) Restarting Apache and PHP 8.1 FPM
root@server:/etc/php/8.1/fpm/pool.d# systemctl restart apache2
root@server:/etc/php/8.1/fpm/pool.d# systemctl restart php8.1-fpm
root@server:/etc/php/8.1/fpm/pool.d# service php8.1-fpm status
…
Active: active (running)
…
Status: “Ready to handle connections”
8°) Checking if the site is back in the web browser
Website is back again !
Thank you all. You contributed to lead me to the solution.