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A double turn will only occur if two types of units participate in the battle, for example, one with an initiative of 16 and the other with 8. Or if one has 12 and the other has 8, then a double turn will occur every other turn (I hope it's clear what I mean by "every other turn"). So, there's nothing complicated about it; you just need to get used to it.
"Wait" is used to wait, as strange as it may seem. Only 50% of the creature's initiative is lost, allowing it to act at a more convenient moment. And it is displayed exactly the same way on the ATB.