Distinction between metric expansion and objects just moving apart from each other? I see explanations likes this:

Galaxies and other sources, then, are not strictly expanding away from each other but rather are attached to the fixed grid on the expanding fabric of spacetime. Thus, the galaxies give us the impression of moving away from each other. (Scientific American)
[...] space is expanding [...] the most correct way to think about it would be that the grid points are getting farther apart. (Physics.se)

How have we determined that it is some underlying grid or space itself that is expanding rather than the objects just moving apart on a static grid of spacetime?
Time will show

...although that is a very long time.

If the grid of the universe was static, objects moving away from us should continue to move away from us with a continues velocity. That is not the case from observed data: Things farther away from us seems to move away faster. This is known as Hubble's law, and is one of the basic concepts of modern cosmology. The model of an expanding metrics fit that data, in that when there is more space fabric between us and the far object, there are more space that stretches, and therefore the velocity is higher.

That things move away from us, and have an accelerating velocity, should suggest a universe that expands faster and faster. There are however other forces limiting this growth, most notably gravity. The fate of the growth of the universe is determined by that unknown balance, that must also consider dark matter and energy.

The magnitude, or even the validity of those models is the current frontier of astronomy, and cosmology in general. Nothing is settled yet.