So in that video when you start the car the trans is in gear and the clutch pedal is depressed, when running the trans is still in gear and the clutch pedal is still depressed, and then when you shut the car off you still have the clutch depressed and the trans is still in gear? Correct me if I have any of that wrong.
If so then your clutch is not disengaging. In that scenario the drive shaft should never spin when starting, running, and when shut off. This is due to the clutch pedal being depressed the whole time and with an properly adjusted clutch the input spline should not turn.