It just needs to be clarified that KL divergence isn’t a proper mathematical norm, so it doesn’t behave the way we intuitively think a distance should. As mentioned, it doesn’t satisfy the triangle inequality, which is a basic property for any distance-like function.
In comparison, both of your examples are much closer to norms as they both satisfy the triangle inequality.
For reference, this is what I’m referring to when I say a “norm”: