schroedinger equation - Where do mass polarization terms come from in many-body Hamiltonian? Why are they sometimes omitted?

This question is about the Hamiltonian for more than one particle (non-relativistic).

Griffiths (Introduction to Quantum Mechanics, 2e) seems to imply that it is $\displaystyle H=-\frac{\hbar^2}{2}\left(\sum_{n=1}^N\frac{1}{m_n}\nabla_{\mathbf{r}_n}^2\right)+V(\mathbf{r}_1,\dots,\mathbf{r}_N,t)$, but wikipedia isn't so clear: https://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29#Many_particles.

Initially the article quotes that formula, but it quickly gets confusing:

However, complications can arise in the many-body problem. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:
$-\frac{\hbar^2}{2M}\nabla_i\cdot\nabla_j$
where M denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as mass polarization terms, and appear in the Hamiltonian of many electron atoms (see below).

Unfortunately, the author wrote nothing below which might explain where the mass polarization terms come from.

Could I get some mathematical background, maybe a derivation, of why these terms are present in the Hamiltonian, and why they are sometimes omitted/forgotten?