I'm assuming one monitor is more tolerant of the poor video sync. than the other.
Check the bottom edge of the tape for any signs of foldover. If present it often shows as a very fine straight line about 0.5mm above the lower tape edge. The control track information is at the bottom edge of the tape, so damage to it could cause those symptoms. If you do find it's damaged, then the tape path needs checking to find why it's not running smoothly within the guide posts.
eg. oxide on guides, worn pinch roller, misalignment of loading arms, etc.
Otherwise, maybe one or both loading arms are not reaching full travel to the end stop.
Another possibility is the video head switching point is out of alignment. There will be either an engineer adjustment on the pcb via a potentiometer or by a button sequence within service mode, for the head switching point). I don't know the model so I don't know how it is in that video recorder.
Cobweb
January 2006