But I'm saying if you evaluate the axiom symbols two different ways, it's two different axioms.
Anyway, this analogy was pushed too far a long time ago.
> I'm saying the meaning behind statements are different.
Yes, exactly! And, just like a single C program can have two different meanings under two different implementations, the same axiomatic system can have two different meanings when deducing its consequences (proving theorems) using different rules of inference.