To me it seems like the tape may either be worn or quite old and that interference or distortion may be triggering the Lite-On to detect the picture as containing copy-protection signalling. An example of such interference could be a buch of lines fluttering through the picture or a wobbly picture for a brief moment.
Just in case it is a bug with the Lite-On firmware, I would recommend at least upgrading the firmware if you have not done so already.
If you have another VCR handy or know someone who has a VCR, you could always try a different VCR, just in case your VCR is triggering the Lite-On to falsely detect the source as copy-protected.
Another option if you are willing to spend a little extra is to obtain a video conditioner. A video conditioner basically re-renders the image by taking the existing image and redoing the signalling between the frames and scanlines. This ensures that a malformed image or an image containing wavy lines will not affect the destination VCR from capturing the image. Unfortunately, as Macrovision is an example of something that deliberately interferes with the image signalling (between each scanline and frame), video conditioners have been taken off the market in some places as these were seen as a way to deliberately encourage piracy.
If you have tried different VCRs, the latest firmware or cannot (or don't intend to) get a video conditioner, you could try using patched firmware at your own risk. Just bewarned that the use of unofficial firmware will likely invalidate any warranty and could potentially harm the recorder.
Finally, just make sure there is no copy-protection on your source tapes. I have encountered a few movies on VHS that were copy-protected despite no mention on the case itself. A simple way to check this is to try and record one of the tapes giving bother onto a 2nd VCR and then play back the recording on the 2nd VCR to see if it recorded fine. If the recording turns out wavy or the picture contrast and brightness starts fluttering, then the source tape is using some sort of copy-protection.
From my experience, my Lite-On LVW-5045 will actually prevent a recording even taking place if the source is copy-protected (such as a DVD), however if I start the recording before I press play on the DVD player, the recording will stop within a few seconds.