I want to thank you all for your help, but it looks like I found the solution, and it's one that makes me feel stupid:
I did NOT (!) know that if you pull the handle down and hold it down, the flush stops earlier (at a higher level). To have a full flush, you have to pull the handle down, but immediately RELEASE it.
OMG! I spent hours looking for the solution, nothing was actually broken and it was so simple.
I'm sorry I bothered you with this.
Thank you!