It is known that there is a $748$-state Turing machine that halts if and only if $mathsf{ZF}$ is inconsistent. So by Gödel’s second incompleteness, $mathsf{ZF}$ cannot find what $S(748)$ exactly is, where $S$ is the maximum shifts function (Also known as the “Frantic Frog”).

I’m rather confused by this fact. As $S$ is well-defined, that should mean that regardless of what axiomatic system we use, the exact value of $S(748)$ always stays the same. We just need a stronger axiomatic system to find $S(748)$, like $mathsf{ZFC}$, $mathsf{ZFC+CH}$, or $mathsf{ZFC+(V=L)}$.

If $mathsf{ZFC}$ and $mathsf{ZF¬C}$ entailed $S(748)$ to be different numbers, since Axiom of Choice is independent to $mathsf{ZF}$, it would be concluded that $mathsf{ZF}$ was inconsistent at the first place. So far, is my understanding correct?