No, you were right to begin with. A 6 hour game has 4 ply (half moves) per day, two for each team. So a vote comes due every 12 hours. Assume the game kicks off at midnight:
0000-0559 White votes
0600-1159 Black votes
1200-1759 White votes
1800-2359 Black votes
Same goes for a 1 day game, you can actually check the game every 2 days. On the opposite days the opponent is voting, and you can't do anything till their vote ends.