For example, if I record a Zoom call with the Xbox program (Windows key + G), can the person see it? Would be for private use only / just for me, of course.
No, because it is an external program and Zoom generally does not tell your conversation partner about this because of privacy.
You can also just ask people beforehand if you can record it before you do.
But I don't think Zoom will respond to third party programs. At most, it is possible that the zoom image can't be recorded at all (similar to Netflix).
You don't see that