Would Bishop Williamson be considered an independent like what was being talked about above regarding independent priests and ABL requiring priests to be part of a religious society if he is going to ordain them?
There is a particular kind of priest that
wants to be independent, like the seminarians refused ordination by ABL. They are happy in their own little world or backyard shed, answerable to no one, isolated from the rest of the Catholic world, no concept of the fight for the Faith. It has the feeling of a little schismatic church. It is not a legitimate canonical situation. Why would
any priest want such an existence?
Then there is the other kind of independent priest who valiantly resisted corrupt authority in the fight for the Faith, and through no desire or fault of his own, ended up 'on the street', cast out of his diocese or religious order. This, I imagine, was the situation of many a faithful priest in the wake of Vatican II. "Pastor Out in the Cold" comes to mind
Pastor Out in the Cold ⋆ Virgo Sacrata Archbishop Lefebvre never required such priests to join the SSPX, which often supported and cooperated in the apostolate with such priests.
Bishop Williamson's situation is similar to the latter, being cast out from the SSPX by the Superior General for his fidelity to the combat for the Faith. He would still consider himself a member of the SSPX in the extraordinary situation of being unable to submit to his superior, preferring true obedience and apparent disobedience to betrayal of the Faith. His situation is a little akin, I think, to Archbishop Lefebvre resigning from his position as SG of the Holy Ghost Fathers, refusing to preside over the destruction of his Congregation in the aftermath of Vatican II, and retiring (or so he thought) to Rome.