I think it’s more semantics than anything, but I don’t think @rmizuno necessarily meant all forms using じゃない are casual, since the first post mentioned only じゃない, the the second indicating the forms that increase formality.
The problem with ないです is that it’s function isn’t strictly within alignment to the simple ない and ます. It projects distality but isn’t used to make the viewer more polite, so the semantics behind formality, distality, and politeness become a sort of mess. For purposes of better meaning, I’m comparing Formal to Informal (Formality), Familiar to Distal (Distality), Casual to Polite (Politeness)
じゃない: Informal, Familiar, Casual
ではない: Informal, Familiar (Forceful), Casual
じゃないです: Informal, Distal, Casual (Semi-Polite)
ではないです: Informal, Distal, Casual (Semi-Polite)
じゃありません: Formal, Familiar, Polite
ではありません: Formal, Distal, Polite
The usage of ない takes the formality out of the statement, while the です adds distance between the participants, at least for the topic of the conversation.