Gaussian weighted distance between pixel amplitudes: motivations and sources?
Maybe not an exact answer, but I'll to give a direction.

What you are using is essentially an RBF-Kernel. First, it has a ready interpretation as a similarity measure (satisfies Mercer's conditions). In essence, other such kernels can be used. For instance, an inner product (linear kernel) would define an angular distance between pixels/features. For a deeper understanding, I would dive in the literature of kernels.

Having said that, the simplest form of such relations are called inverse distance weighting (Grady could have used this as well), where any identity like $1/d(g_i, g_j)^p$ can be used. These methods also cover a wide spectrum.