The accumulated elevation gain between two points must *not* be path dependent, or else you could go up the climb the "short" path and down the "long" path and gain more energy going down than you use going up, and that violates one of the laws of thermodynamics because it's a recipe for a perpetual energy machine. But I agree you get different results if you measure with a different ruler -- that's because there's a difference between a distance and the measurement of a distance.

And the problem of interest to cyclists isn't quite accumulated elevation gain, it's the energy they will need to climb whatever that accumulated elevation gain is, or perhaps the energy per unit time. Because traditionally cyclists didn't have a way to measure or estimate that energy, they got used to a proxy: elevation gained (although that doesn't exactly describe the energy problem because total elevation gained doesn't tell you how steep the road was). Fortunately, we now have devices that allow us to measure that directly, viz., power meters. So that's what I use: I translate the energy expended into an equivalent amount of climbing. I could have called that equivalent elevation but instead I called it virtual elevation. I actually calculate the virtual gradients for each second of the ride and combine it with the speeds for each second of the ride to get cumulated virtual elevation -- because I do it that way, I get not only the total elevation gained but also its distribution along the way, i.e., the virtual road profile.