How can different voltages produce the same short circuit current?
In the video, the power supply can provide a maximum of 150 amperes.  

If the resistance of the short is low enough, then more than 150 amperes would flow.  

The cables used in the video are very thick and so have a very low resistance, and would allow more than 150 amperes of current to flow.

But, the power supply can only provide 150 amperes so you only ever see it  putting out 150 amperes.

Take a couple of examples:

Set the power supply to 1 volt.  Assume a cable resistance of 0.005 ohms.  That would be a current of 200 amperes.  The power supply can only deliver 150 amperes, so that's all we see in the video.

Now set the power supply to 20 volts, but keep the cables with 0.005 ohms.  Ohm's law says you should get a current of 4000 amperes.  The power supply can still only deliver 150 amperes, so that's all you will get.