OK, I just took another look at that thread. As I suspected you properly corrected me, but never were able to give an alternative way of expressing it purely in non-mathematical terms. That’s why I put in the “manifestations of the same underlying thing” clause.
But if even that is not a good enough way to express it, can we try to come up with a phrase that satisfies you and still addresses “What are these units that can be manifested by either mass or energy?”
Your version there was “a system that has energy of E and a mass of m.” and “Imo it’s much clearer to say that the mass of a system (if you’re consistent in defining the system) is absolutely conserved. However one form of the system’s energy can be converted to another, and the individual particles that make up the system can change resulting in a local mass defect.”
That’s technical English but not popular science English. And one is convertible into the other. 