At one point I do believe williamsii was in the genus ariocarpus
Here's a list of the names it's had in the past with the person who named it to the right,
Lophophora williamsii - (Lem. ex Salm-Dyck) J.M.Coult.
Anhalonium visnagra - K.Schum.
Anhalonium williamsii - (Salm-Dyck) Rümpler
Anhalonium williamsii - Lem. ex Förster
Ariocarpus williamsii - (Salm-Dyck) Voss
Echinocactus lewinii - (Henn. ex Lewin) K.Schum. in Engl. & Prantl
Echinocactus williamsii - Lem. ex Salm-Dyck
Echinocactus williamsii var. anhaloninicus - K.Schum.
Echinocactus williamsii var. pellotinicus - (K.Schum.) Rouhier
Mammillaria williamsii - (Lem.) J.M.Coult.
3... 2... 1... BLAST OFF!!!!FFO TSALB ...1 ...2 ...3
My grafting guide