After some trial and error, I found out, that when I deactivated and then reactivated the device in the device manager, it would work perfectly normal over DVI. So, as my display has both DVI and VGA inputs, I decided to connect it via both (my PC has both outputs) and see if I could get it to work. While thinking about it, I remembered that when I first installed the driver, it was working with the Intel driver active until the next reboot. When this happened to me, I was quite disappointed first, but then I started to thinks of ways to solve the problem.
As many others here, I am affected by the problem that the Intel driver for the integrated graphics doesn't work when a display is connected via DVI or HDMI (VGA works fine).