We work in the setting of linear skew constacyclic codes over a commutative base ring $S$. We show that the notions of $(n,σ)$-isometry and $(n,σ)$-equivalence introduced by Ou-azzou et al coincide for most skew $(σ,a)$-constacyclic codes of length $n$, and moreover identify the classes for which the two notions can differ. To prove these results, we first determine all Hamming-weight preserving isomorphisms between their ambient Petit rings which extend some automorphism $τ$ of $S$ that commutes with $σ$. We then prove that when those ambient rings are not associative, these isomorphisms must have degree one, and give examples of non-degree one isomorphisms in the associative case. As a consequence, we provide new definitions of equivalence and isometry that for skew constacyclic codes over finite rings exactly capture all Hamming-weight preserving isomorphisms between their ambient rings, leading to tighter classifications.

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