Select with limit, get total count, optimize query
MyISAM or InnoDB?

Your link does not say anything about SQL_CALC_FOUND_ROWS being slower.  And I would expect it to be faster if for no other reason than having one fewer round trip to the server.

The risk comes when you have thousands of rows.  Then either method may be spending more time than you can afford.  Come back when you encounter that; we can discuss workarounds.

Hmmm... I wonder if the LEFT makes the CALC run faster, since it should not have to reach into users after 12 rows.  But I fear that the optimizer is not that smart.  I'll guess that the following subquery will avoid that issue:

SELECT  SQL_CALC_FOUND_ROWS mr.*, 
      ( SELECT  name
            FROM  users
            WHERE  id = mr.user_id 
      ) AS name
    FROM  medias_ratings mr
    WHERE  mr.id_media = XXX  -- changed by EDIT
      AND  mr.id_client = YYY
    ORDER BY  mr.`date` DESC
    LIMIT  12;


Now, for indexes.

mr:  INDEX(id_media, id_client, date)  (Put the '=' columns first, then the ORDER BY.) -- changed by EDIT

users: Add a 'covering' index: INDEX(id, name) specifically to make this query faster -- it can run "Using index", as seen in the EXPLAIN.