How do I take optically faithful videos of glow props in the dark?
Deciding whether or not to shoot interlaced is really a matter of matching your destination format.  If you're shooting video for the web, you should record it in progressive scan mode.  You can record interlaced, and then de-interlace it in software, but this will likely lead to video artifacts, especially considering your subject matter.

I think the picture you posted has misled many commenters into giving you the wrong advice.  Because the picture is blurry, they recommend a faster shutter speed, which will sharpen a moving object.  But, given the nature of the video you linked to, I think more motion blur will result in a smoother, more natural look.  So try recording progressive, with a slow shutter speed.  This will let more light into the lens, brightening the exposure and making longer motion trails.

Also, try to match the framerate with which you're recording to that of your destination.  If you are recording at 29.97 and vimeo or youtube is playing at 24fps, they're going to introduce some frame blending on their end, which you have no control over, and which is likely to introduce artifacts.