I was getting the same error as well "ERROR 2002 (HY000): Can't connect to local MySQL server through socket '/var/lib /mysql/mysql.sock' (111)". I have uninstalled and re-installed MySQl several times and have spent multiple hours troubleshooting the issue. The issue that I notice is when I added the credentials in the my.cnf option file located in /etc/my.cnf as this will cause MySQL issues with starting the services and would throw the error.
To resolve this, I removed the credentials out of the my.cnf option file in /etc/my.cnf and I create a .my.cnf option file in the home directory. For example /home/username/.my.cnf in Linux (replace username with your username in Linux) and add the credentials in there. No issues after that.