Imagine that you stretch a perfect spring out to a given length. The total energy input into the system is work = force*displacement. If this is a perfect spring, this is the energy stored in the system, when you let go, the spring will bounce forever. Now imagine an imperfect spring, the spring's damping dissipates energy when you release it, and eventually after a certain number of cycles it will stop. The energy stored in the system is the initial work you put in, and the energy dissipated is the energy lost when the spring cycles. This ratio between them is damping.

There are many ways to define and measure damping. The wikipedia definition you are looking at is not very intuitive for electrical systems in my opinion, and I feel like you are extrapolating information.

A highly resonant second order system (low damping or high q) will not have any amplification of displacement for an impulsive signal. For a lightly damped system, with a steady state signal, energy can accumulate in the system over time resulting in amplification of the displacement. If there is no damping, the system will be unstable and blow up, otherwise due to losses there will be a finite amount of gain.

Consider taking a control systems class, or looking for lecture notes online. Often they start with a foundation of first and second order systems, and you can learn about all the ways to measure and characterize their performance (settling time, overshoot, step responses, impulse responses, bode plots, etc). Sometimes in speaker land we forget that just looking at frequency magnitude doesn't exactly describe what is a system is doing in time.