Interpretations are designed to give the same predictions as can be inferred from a no-interpretation math, otherwise they would be called theories.

This isn't necessarily true. Consider that the GRW interpretation has been pretty much falsified by Van Harlingen's work at UIUC (macroscopic current superposition in SQUIDs). Most of the interpretations rely on different postulates than traditional Copenhagen quantum so there can be (and generally are) differences. However, to date, most of those differences aren't measurable.

Similarly, we call many-worlds an "interpretation" even though no one has figured out how to actually make predictions with it. The difference between "interpretation" and "theory" is a bit loose.

Interpretations are designed to give the same predictions as can be inferred from a no-interpretation math, otherwise they would be called theories.

This is empirically false, as a statement about how scientific discourse works. Compare string theory, which is frequently labeled a 'theory' (or family of theories) even though it has far more difficult-to-observe posits (one-dimensional strings!) than most (perhaps all) of the mainstream QM 'interpretations'. See also falsified QM interpretations.

Perhaps a more model-theoretic approach would be appropriate here; clearly QM interpretations can vary quite a bit in their verifiability/falsifiability, so what distinguishes them from other theories may be that they specify the meanings of the terms in the QM formalism. On this view, 'interpretations' may add real content and predictions to a set of statements, provided that in the process they also fix the semantics of a large portion of the statements. After all, the problem with QM is not merely that we aren't clear on the invisible metaphysics secretly underwriting and accounting for our experiences; we aren't even clear on the phenomenology (appearances) or ontology (observable posits) of the theory, treated as mere formalism.