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