You need to continue using a 9.4. User-Defined Variables, not a 13.6.4.1. Local Variable DECLARE Syntax:
BEGIN
-- DECLARE variable INT DEFAULT 0;
SELECT rank, best_time, user_id
FROM (
-- SELECT SET variable = variable + 1 AS rank, best_time, user_id
SELECT @variable := @variable + 1 AS rank, best_time, user_id
FROM database.user_round, (SELECT @variable := 0) init
WHERE round_id = 1
ORDER BY best_time ASC
) AS result
WHERE user_id = 1;
END