Skud Pai Sho and "Avatar"

[Awyrlas]

Are the rules making sense throughout the series? Because sometimes it's shown the game is played with tiled placed in the squares, while sometimes on line intersections (there are some snapshots in the Avatar wiki on Fandom). Is the White Lotus opening (done by the members of the White Lotus order as a secret greeting to identify members) making sense in Skud Pai Sho?

In the show, they do favor playing in the boxes, though the inconsistent portrayal of the game up until Legend of Korra meant that early fans of Pai Sho settled on playing on the intersections. This was likely chosen due to its visual similarity with other games that inspired the in-universe depiction of Pai Sho like Go, Shogi, and Xiangqi. Ever since then it's largely stuck, though some other variants opt to play in the boxes. 

This is certainly not stopping you from doing so yourself, however. I think Skud Pai Sho would be more than playable within the boxes, though much different than the game that the community has been playing for some time.

Executing the White Lotus Gambit would not be possible using the rules of Skud Pai Sho, though its likely that the White Lotus Gambit was closer to a secret handshake than it was a real strategy employed by players of the game.