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| A cistern is simply a word a tank that contains water or other fluid.... so yes, what you've posted a photo of is a cistern... and it's the same in german; zisterne, italian; cisterna | |
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Almost true

In Italy "cisterna" describes some large water reservoir, like the emergency supply for a house or a shelter; the one we are talking about is called "cassetta dell'acqua" (literally "small box for water") or simply "cassetta".
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| Underneath the floating device is a rubber seal that stops water running into the toilet bowl. | |
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Should this not work, the seal can easily be replaced with a new one. It can probably be found in stores like Jumbo / Obi etc. The size should be standard, but remember to bring the sample just in case.
The typical effect of an incomplete seal is a small but steady flow coming down. It starts as something quite small that you don't notice, then you begin hearing it at night when there's silence, finally it becomes apparent. Apart from the waste of water, with time it might also create yellow/orange calcium deposit inside the bowl.
Two years ago it happened to me, I dedicated one Saturday morning to a complete disassembling and cleaning of the moving parts and the interior of the tank. My model had two buttons, one for partial and one for complete discharge and they are linked to two separate cylindrical mechanisms, one inside the other.
Simple machine, but quite funny to understand. And testing it once reassembled brought me back to when I used to play with water in the childhood