I've been testing this for the past two days and now LCD worked fine everytime after first reset. The point is, if arduino is not connected to the power source, the screen usually doesn't work when I plug it in. After first reset, it is back to normal. I have to check what happens when it is turned on for a longer time.
Regarding your post, I would say there could be something with the init sequence...
LCD is in serial mode.
here you can see, what happens after powering the arduino on. Sometimes these characters are covering all the screen, not just part of it...