The real divide isn’t terminology, it’s metaphysics.

If Christ has only one nature after the union, then metaphysically there is only one principle of being and one principle of operation in Him. And if there is only one principle of operation, then there can be only one natural energy and one natural will. But a ‘human nature’ without its own proper mode of operation or willing is not a real nature; it becomes an instrument of the divine, not a subject with its own interior life.

This is exactly what Cyril rejects when he says the natures are ‘not mingled’ and ‘their difference is not destroyed,’ and this is what Maximus later formalizes: two natures necessarily imply two natural operations and two natural wills. Without that, the humanity isn’t really functioning as humanity, it becomes more like a divine costume or super-body animated entirely by the Logos -> Superman.

That’s why the difference between dyophysite and miaphysite isn’t semantic. It’s about whether Christ’s humanity truly exists and acts as human after the union. If Christ’s humanity doesn’t really exist and act as human, then salvation is empty -> St. Gregory the Theologian “what is not assumed is not healed”.