They're called FLOORS!

[Tsukino]

I wouldn't be so quick to point fingers. You should take into consideration what the players call them.

Example: 真成編１層、真成編２層、真成編３層、etc.

How would you, as a player, refer to each 層 in English? A turn? A floor?

層 following a number absolutely means "floor," though it is obviously less common than 階. The localization team was probably trying to go for something similar with their naming - i.e. more unusual/exotic than "floor."