The NIH website refers the two substances listed as Hydrotreated which according to the to the Machinery lubrication website are part of Group II. Are Group III's supposed to be Hydrocracked will the CAS registry specifically say hydrocracked?
As always in these numerous and repeat threads on base stocks, it is important to remember that the base stock Group designations are not intended to describe or delineate finished oils but to provide interchange information for manufacturers and blenders. A good place to start is API 1509 Appendix E which is referenced in the
Machinery Lubrication article you link.
https://www.api.org/~/media/Files/Certification/Engine-Oil-Diesel/Publications/AnnE-REV-09-20-19.pdf
API notes that there is not only one way to meet the requirements of a particular Group, especially with Groups I--III:
Base stocks may be manufactured using a variety of different processes including but not limited to distillation, solvent refining, hydrogen processing, oligomerization, esterification, and rerefining.
So as you can see, a Group III base stock is not defined by the method of production but rather by distinct physical properties that are detailed in API 1509 Appendix E. The remainder of the Group designations (IV and V) are described in the same document but are of a different nature than the first three Group types.
As an aside the way you determine the performance of a formulated motor oil is by the specifications, approvals and certifications the oil meets. Base stock may contribute to all of those but it is not a substitute.