What you're saying is more or less equivalent to “two C implementations targeting different [architectures / operating systems / whatever] are actually implementations of two different programming languages”.
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.