Adding on to this…

The “rest mass” is so important to physicists because it is an invariant quantity under relativistic transformations. Think about kinetic energy. Kinetic energy is a form of energy, and therefore a form of mass. But different observers at different velocities can look at the same object and come up with different kinetic energies for it. But “rest mass,” the mass it would have at rest, is something everyone can agree on.

Finally, as the battery on the scale example demonstrates, photons make their own gravitational field from their energy. Since photons make gravity, they can (theoretically) make black holes. "Kugelblitz" is the name for a black hole that has been composed of only photons.