Before you added the second part - this sounded exactly like the problem I have. From experience I know it's the rubber seal at the bottom of the cistern which isn't sealing the hole - hence water leaks out - hence comes into the cistern to fill up to the amount required to turn the cistern off. If your problem is at the top, this answer may not help. - Something to consider though.
How to get a replacement part is another problem - that's mine - no-one seems to sell cistern rubber seals, most prefer to sell you a new unit instead. You can get them though.
Jaki
January 2009