First about the terms:
- The outer side of the cranial bones is covered by a single layer of periosteum that is specifically called pericranium.
- The inner side of the cranial bones is covered by dura mater that has 2 layers:
- the periosteal (or sometimes called endosteal) layer that is attached to the inner side of the bones
- the meningeal layer that covers the periosteal layer and, at places, folds into the skull to make extensions
Image 1: Dura mater with its periosteal and meningeal layer (source: Idris Siddiqui, SlideShare)
Image 2: Extensions of the meningeal dura mater (falx cerebri, falx cerebelli, tentorium cerebelli) (source: Wikipedia, CC license)
The cranial dura mater has the periosteal and meningeal layer, but the spinal dura mater has only a meningeal layer, which encloses the spinal cord. Vertebra have no additional periosteal layer. The periosteal layer of the dura mater that covers the cranial bones does not extend to the spinal canal:
The periosteal or endosteal layer of dura mater is simply a layer of
periosteum that covers the inner surface of the skull. The layer does
not extend beyond the foramen magnum to become contiguous with the
dura mater of the spinal cord. The spinal cord dura mater has no
The meningeal layer of dura mater is a durable, dense fibrous membrane
which passes through the foramen magnum and is continuous with the
dura mater of the spinal cord. (V. Kekere, Dura mater, StatPearls)
Both quotes from the book mentioned in the question suggest that the periosteum and dura are two separate things: periosteum being a part of the bone (as elsewhere in the skeleton) and dura consisting of a single layer. The other explanation, which I presented here, is that the periosteum on the inner side of the skull is a part of the dura (its periosteal layer) that has another layer (the meningeal layer). So, they are two ways how you can look at the structure of dura mater, and I believe, the second one is more widely accepted.