I believe there is a distinction between those concepts.

Axioms are true by their own definition. Therefore, discovering an axiom to be false is a concept that is inherently illogical.

Discovering that a formal verification system produced an incorrect output due to a bug in its implementation is a perfectly well-defined concept and doesn't led to any logical contradictions; unless you axiomatically define the output of formal verification systems to be necessarily correct.

I believe this definition don't make sense in the general case considering the number of vectors that can introduce errors into software.