Comment by neutronicus

A simple, classical case where "stiff" fields arise is electrostatic plasma physics.

Electrons are tiny and nuclei are huge, so you have a bunch of mobile charge carriers which cost energy to displace to an equilibrium position away from their immobile "homes". A collection of test charges moving "slowly" through the plasma (not inducing B field, electrons have time to reach equilibrium positions) will produce exponentially decaying potentials like in the article. If you want to read more, this concept is called "Debye Screening"

Anyway, this might be a more helpful approach than trying to imagine a spring - a "stiff" field equation is an equation for a field in a medium that polarizes to oppose it, and you can think of space as polarizing to oppose the existence of a Z Boson in a way that it doesn't polarize to oppose the existence of a photon.