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.
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.