Nondeterministic polynomial time algorithm versus certificate/verifier for showing membership in NP
The certificate you propose might not be polynomial in the size of the input.
The input size of the problem is $O(n^3 + \log k)$, while your certificate has size $\Omega(k \log n)$. This is not polynomial, e.g., for $k = 2^n$.
Your certificate would work if you set it, e.g., to an empty list whenever $k = \Omega(\frac{n^2}{\log n})$, and modify the verifier to just check whether the input configuration of the cube is solvable for any such value of $k$ (thus ignoring the certificate).