Just becuase the issue has only shown up with the video camera does not allow you to determine if the camera has poor immunity or the lathe hs excessive emissions. It could be either, or a bit of both.
As Andrew has already said the VFD unit is a component intended for industrial applications Thus it is designed to industrial emissions regulation maximum levels which are higher than domestic. Assuming the camera is a domestic unit it legally only needs to be immune (perform to a level defined by the manufacturer) to the lower domestic levels of interference.
It is actually quite difficult to get most VFD emissions down to the domestic levels. I've done it on various equipment. An example was bench mounted machine with three small drives and had to run from a "13A" socke in a lab envvironment. It required additional filters on the input of each drive, reactors (inductors) on the outputs, putting the drives inside a screened box (in addition to the basic metallic enclosure of the machine) and screened cables.
To reduce leakage current from the filters and screened cables an isolation transformer was required (This had some tricks in it's specification that helped with other issues). Adding the isolation transformer then required extra attention to electric shock protection…….
That level of effort is OK for a professonal machine costing 5 to 6 figures. People selling 3 to 4 figure hobby tools tend not to bother so much.
Vic, If you are anywhere near Cambridge (UK) I could come round and make some measurements.
Robert G8RPI.