Vegemite, I know this is difficult and frustrating for you, and I am sorry for that.
What you're encountering isn't unusual, and based on what you're telling me I don't think the school (for once) is intending any deception.
Pay is tied to classes. When classes end, so does pay. However, a lot of schools will extend the contract period a bit beyond the end of classes. This is done to give you a chance to move, travel, transfer your residence permit, or whatever it is you're going to do before you either move on, start a new year, or return home. This post-class grace period, while unpaid, is actually kinda meant as a courtesy to you.
Of course, communication is never very good in most schools. Of course it wouldn't occur to you to ask if that last period was paid. And it would definitely never occur to the school to explain it to you.
Again, based on what you're saying, I genuinely don't think the school is setting out to cheat or deceive you. And I don't think there will be much you can do about it. It's something you're going to have to write off as "cross-cultural education."