Evie, I see your point.
Many fitness professionals include myself define the core as everything between the head arms and legs as you have stated and I include myself among the many. However, if we are going to define it according to the textbook, the shoulder girdle is not inclusive of it. This is where I stand corrected.
When I am training my clients and I observe that the core in the pure definition of core is weak, there is generally weakness and instability in the scapular. Hence, I like many other fitness professionals design programs to improve core stability and strength/endurance as well as the stability and strength of the shoulder.
Thank you so much for your thought-provoking question and best to you!