Mail time: what system do I use for vertical springs?

A question from Hibisca, who attended my Atlanta summer institute last year:

I have a question about approaching energy with vertical mass-spring systems. I recently made the mistake of teaching my students to include gravitational potential energy in the analysis after choosing the new equilibrium position as the location where the spring hangs at rest with the mass attached. I have since mathematically proved to myself why PEg should be left out (and this Khan Academy videogoes through the step-by-step proof as well), but I have not found a clear conceptual explanation. If we analyze the block-spring-Earth system, there is no external work done on the system, mechanical energy is conserved, and it seems that PEg should be included. How do you explain this to your students? I assume relying on a mathematical proof is not sufficient.

This one's rather complicated. I always suggest, unless you're told otherwise, to treat a vertical spring as the spring-object-earth system. Then there are no forces at all external to the system, so mechanical energy is conserved.

The SYSTEM potential energy is just 1/2kx^2... where x is measured from the equilibrium position, not the position where the spring is unstretched.

See, you could include both a separate spring potential and a gravitational potential if you measure everything from the unstretched position. But that's really complicated. Why not fold all the potential energy into a single term? You can show mathematically -- and I suspect that Khan Academy does show -- that the 1/2kx^2 potential energy with x measured from equilibrium is entirely equivalent to both 1/2kx^2 and mgh terms measured from the unstretched position. But I don't think that's important. Just use the single term, and recognize that it's for the spring-object-earth combined system.