The concept of a screening mass is a powerful tool to simplify the intricate physics of in-medium test charges surrounded by light charge carriers. While it has been successfully used to describe electromagnetic properties, its definition and computation in QCD is plagued by questions of gauge invariance and the presence of non-perturbative contributions from the magnetic sector. Here we present a recent alternative definition of a gauge invariant Debye mass parameter following closely the original idea of Debye and H\"uckel. Our test charges are a static heavy quark-antiquark pair whose complex potential and its in-medium modification can be extracted using lattice QCD. By combining in a generalized Gauss-Law the non-perturbative aspects of quark binding with a perturbative ansatz for the medium effects, we succeed to describe the lattice values of the potential with a single temperature dependent parameter, in turn identified with a Debye mass. We find that its behavior, as evaluated in a recent quenched lattice QCD study, deviates from that in other approaches, such as hard-thermal-loop perturbation theory or from electric field correlators on the lattice. In particular around the phase transition its values tend to zero significantly faster than at weak-coupling.