Almost all "interpretations" (using the word as used in the literature) of quantum mechanics use different axioms, and its a mathematical question as to whether or not the theories make the same predictions. Many stochastic "interpretations" modify the Schroedinger equation, for instance. Even many-worlds can't be proven as an interpretation using your definition (no one has shown it actually leads to the same predictions as Copenhagen).

Its an unfortunate artifact of the literature on various approaches quantum mechanics that the words interpretation and theory often over-load each other, but its the reality we live in.