Also there is a Timing Diagramm printed, but no exactly times mentionned...
The times are given in the datasheet for the controller, but not always in the datasheet for the module. If you don't have the specific datasheet for your controller then use the one for the 44780 (www.sparkfun.com/datasheets/LCD/HD44780.pdf).
So what do you think? What do I have to change in the liquidcrystal library to may make it work?
Nothing. You might want to try downloading a new copy of the Arduino IDE which will include a new copy of the library. We are really grasping at straws here but it's worth a try. You will probably wind up needing another LCD module, from a different supplier, to pin down the problem.
Don