Product review (3 stars): I think I got a permanent fix!
This is the third one of these for this fridge.  The "spring" it uses to return to its original position when you stop filling a water glass is the rubber of the pad itself.  After a few years of doing this, the rubber stops being so... well, springy.  This leads to the water or ice NOT turning off when you remove your glass.  When a small child does this, and is alone in the kitchen, they can think nothing of it, and just walk away.

Messy.

I had some springs of the appropriate size out in the garage.  I put one on this to assist the rubber, and I believe it is the last one of these I will ever need.  No more mess.