AUGGGHHHHHHH!!!!! This video has the horrible advise of a half credit card! USCutter, get rid of this video!!! An why tell the proper way to do it using vinyl then tell a way to do it using paper?
Ok, now that that is over, Yes, that is a grounding wire, and what he showed doing is completely safe. the cutter is not plugged in at this point, so there is no power to spark to anything anyway. Once the cutter is plugged in, this is simply a ground wire, if there is ever a spark on a ground wire, something is very seriously wrong with either the machine or the wiring of the house/building.
So other than the setting blade depth thing, what about the video is confusing?