For this achievement, you need to win a game in every game mode. The set of pieces you are using doesn't matter. The puzzles and challenges do NOT count towards this achievement either.
These are the modes that I had to play before the achievement popped. They were NOT done in this order.
Single Classic Chess - Normal Placement
Single Classic Chess - Random Placement
Single Battlegrounds Duel
Single Battlegrounds Slasher
Campaign Order Campaign
Campaign Chaos Campaign
Single Minigames One-timers Menace
Single Minigames One-timers Execution
Single Classic Chess - Tactical Placement - The achievement popped when I won a game on this mode.
If people can tell me what they played and when they got the achievement, I can hopefully narrow down this list :)