mysqli::mysqli(): (HY000/2002): Can't connect to local MySQL server through socket 'MySQL' (2)

When you use just "localhost" the MySQL client library tries to use a Unix domain socket for the connection instead of a TCP/IP connection. The error is telling you that the socket, called MySQL, cannot be used to make the connection, probably because it does not exist (error number 2).

From the MySQL Documentation:

On Unix, MySQL programs treat the host name localhost specially, in a way that is likely different from what you expect compared to other network-based programs. For connections to localhost, MySQL programs attempt to connect to the local server by using a Unix socket file. This occurs even if a --port or -P option is given to specify a port number. To ensure that the client makes a TCP/IP connection to the local server, use --host or -h to specify a host name value of 127.0.0.1, or the IP address or name of the local server. You can also specify the connection protocol explicitly, even for localhost, by using the --protocol=TCP option.

There are a few ways to solve this problem.

1. You can just use TCP/IP instead of the Unix socket. You would do this by using 127.0.0.1 instead of localhost when you connect. The Unix socket might by faster and safer to use, though.
2. You can change the socket in php.ini: open the MySQL configuration file my.cnf to find where MySQL creates the socket, and set PHP's mysqli.default_socket to that path. On my system it's /var/run/mysqld/mysqld.sock.

3. Configure the socket directly in the PHP script when opening the connection. For example:

   $db = new MySQLi('localhost', 'kamil', '***', '', 0, 
                                 '/var/run/mysqld/mysqld.sock')
   

If it's a PHP issue, you could simply alter the configuration file php.ini wherever it's located and update the settings for PORT/SOCKET-PATH etc to make it connect to the server.

In my case, I opened the file php.ini and did

mysql.default_socket = /var/run/mysqld/mysqld.sock
mysqli.default_socket = /var/run/mysqld/mysqld.sock

And it worked straight away. I have to admit, I took hint from the accepted answer by @Joni