To give a realistic impression, I wanted to show my fingers too, not just the device screen. One other way I could think of is to use two phones, one for running the app to demo, the other shooting the video and projecting it via the HDMI cable... But it's a bit difficult to setup - switching projector cables is often quite painful.

The problem with my original approach is that I couldn't find a good webcam that records videos as well as the phone.