Connect the oscilloscope to equipment with a common neutral and ground
Say that you incorrectly clamp the GND of the probe in the T1 of your "DUT", if your oscilloscope is floating, then all its grounded metals and shielding become a direct contact to the live wire. And you can get "tickled", as you said yourself.

In the picture you drew the GND probe was disconnected from the GND/Neutral conductor, but the measuring probe is still connected to the live wire. In that case, your oscilloscope will be referenced to the live voltage (if you measure the voltage with a multimeter from ground (earth) to any point on the oscilloscope circuit it will be "about the same" as the live wire). 

I think your confusion (at least looking at that ball with L G N) stems from you thinking the GND of the oscilloscope is connected to the GND of the house wiring, which in that video is not the case (the oscilloscope is floating, no connection to the power outlet ground pin).