OK, here is what you want (one line change)?
SELECT @seq := 0, -- raw 1,2,3,...
@rank := 0, -- rank, allowing for ties
@prev := 999999999; -- for catching ties
SELECT Rank, gamer, point
FROM (
SELECT @seq := IF(@prev = point, @seq, @seq + 1) AS seq, ## CHANGED
@rank := IF(@prev = point, @rank, @seq) AS Rank,
@prev := point as prev, # I forgot this
gamer,
point
FROM (
SELECT gamer, SUM(score) as point
FROM score_ranked s
GROUP BY gamer
ORDER BY point DESC
) x
) y
WHERE Rank <= 10
ORDER BY Rank, gamer;
+------+-------+-------+
| Rank | gamer | point |
+------+-------+-------+
| 1 | Alp | 123 |
| 2 | Ben | 111 |
| 2 | Bob | 111 |
| 3 | Dan | 77 |
| 4 | Ira | 66 |
| 5 | Jay | 55 |
| 5 | Jil | 55 |
| 5 | Jon | 55 |
| 6 | May | 44 |
| 7 | Wes | 33 |
| 7 | Won | 33 |
| 8 | Zoe | 11 |
+------+-------+-------+
(Probably @seq and @rank could be collapsed into one variable.)