This all makes sense to me. I think one remaining question here is if we have any definitions of "identical" and "interchangeable" that are independent of each other. Do we believe there is ever a representation that is identical and is not interchangeable? Or is this an "if and only if" relationship where identical and interchangeable are just two different words that mean the same thing?
If a representation a is always interchangeable with itself — which sounds like a reasonable assumption — and we have the if and only if relationship back to identical, I think we then have the axiom of Reflexivity on identical where a representation a is always identical to itself. Correct?
What about going from a representation being identical to a also being a copy of a? Was that also an if and only if relationship such that all identical representations are also legit copies?