Camera not detected in Skype and Zoom, but works fine with other Apps

First things first: I am not frustrated, I am really ******! Pardon my french. What happend to Macs that just simply worked? Sorry, now for the facts:

I am using a Mac Mini (2018). As I had to setup two more worksstations for my kids due to the corona crisis I needed another webcam for my Mac Mini. There are lots of "real" Cameras (DSLR, Mirrorless...) within my household I tried to hook one up via USB. For sure no success. As there is actually no way get a webcam I bought an hdmi video capture device (BlackMagic UltraStudio Mini Recorder). I won't get into details about getting it to work with Mac OS - it's a pain in the... Just this: Till now I have spend more than 200,- $ only on cables...

Finally I got the setup to work - almost. The camera is detected by the BlackMagic software, I can record video, it works with google duo, with microsoft teams, with quicktime, but not with Skype, FaceTime nor Zoom.

Within Zoom, Skype and FaceTime the camera is not recognised ("No camera connected"). Does every Software on Mac OS have its own camera drivers?

How can I make it work with those Apps?

Thanks,

Sven