I'm not happy with the notion that something could "be" metaphysical instead of natural. We can propose metaphysical explanations for phenomena that don't have a physical explanation (yet), such as lightning, but it's only the explanation that is either physical or metaphysical. The phenomenon itself just "is".
In the discpline of Philosphy you are correct, however I think it is being used in Schroeders book as per the definition, ie supernatural.
Of course the problem with "supernatural" explanations is that they're so often made redundant by natural explanations - e.g. lightning. I think we should regard "supernatural" as something like "ultraviolet" - i.e. "beyond visible light, "more" than visible light in the sense that it's more energetic, but not fundamentally different from visible light.