Since there are computable total functions that are not primitive recursive, one cannot make the two notions of time complexity coincide. If we add any primitive recursive function as an initial function in the primitive recursive hierarchy, the resulting hierarchy will still consist entirely of primitive recursive functions. And so we may take any computable total function that is not primitive recursive, and this function can have no "recursive definition" as a primitive recursive function and thus has no Gödel complexity.

Since there are computable total functions that are not primitive recursive, one cannot make the two notions of time complexity coincide. If we add any primitive recursive function as an initial function in the primitive recursive hierarchy, the resulting hierarchy will still consist entirely of primitive recursive functions. And so we may take any computable total function that is not primitive recursive, and this function can have no "recursive definition" as a primitive recursive function and thus has no Gödel complexity.

Furthermore, if one entertains the idea of addressing this issue by adding a computable function $g$ that is not primitive recursive, and building the primitive recursive hierarchy on top of that function, then again it will not succeed, since the class of computable total functions is not the same as those that are primitive recursive relative to any fixed computable total function $g$. One can prove this by observing that we have a computable function that is universal for all such functions, simply by unwrapping the primitive recursive definitions and evaluating them. And so by diagonalization there will be a computable total function that is different from any function obtainable by performing primitive recursion over $g$.