You're right, it's not arbitrary for that purpose. I was trying/hoping to avoid that subtlety because much of the recent discussion has been about how the rest energy changes as energy enters and leaves the system; and because (as you pointed out while I was writing this) we can jus weigh the system to establish its rest energy.Is this convention really arbitrary in the context of reckoning a system's rest energy?
What contributes to a system's rest energy? Answer: all of the energy "internal" to the system.
The potential-energy contributions associated with the relative positions of a system's constituents can be positive or negative, depending on whether the force in question is repulsive (positive) or attractive (negative). The potential-energy contributions approach zero in the limit that the system's constituents are infinitely far apart.
Since we're having this discussion in the relativity forum (as opposed to the classical physics forum), it's worth pointing out that, in GR, all of these statements have limitations.(or we can just weigh it)