Marking certain details by hand is likely to be the most common way of doing it.
I'm not sure but I'd think this very rare tool was meant for workers who did this most of the time. To make this tool work properly you have to do it on a thick but soft piece of felt or similar material, like shown in the picture of the original post! This material is meant to "catch" the stitches and create a loop when the cloth is removed from the mat.
Cutting the stitches is still to be done with scissors by hand!
These are the pictures from the instruction flyer:
But I think the OP wanted to know if there is a special term for this procedure.
Well, in german it's called Stiche einziehen" oder auch "Stiche durchschlagen", no matter how it is done, by hand with a needle or with a tool similar to those shown!
SG