I really do not know what this really is about, but someone in the reversi board proposed a game called "reversi race", where you count the total number of disc flips throughout the game instead of the number of black/white disks in the end.

Unfortunately, that proposer ignored the fact that if no-one passes during the game and the whole board is filled, the same player that flips more disks also has more disks in the end, so the difference between this variant and ordinary reversi is very small. Actually, we have the formulas:

BS - WS - WP + BP = 2 (BF - WF),

assuming that the total number of moves (including passes) in the game is even, and

BS - WS - WP + BP + 1 = 2 (BF - WF),

if the total number of moves (including passes) is odd.

Here BS is black's score in normal othello, BP the number of black's passes, BF the total number of disks flipped by black, and Wx similarly for white. We assume that white plays first.

IMO reversi with the tournament rules (fixed diagonal opening, black moving first) would be a far more important thing to implement.