According to the Pope's latest Motu Proprio, "On the Service of Charity"
A charitable agency may use the name “Catholic” only with the written consent of the competent authority, as laid down by canon 300 CIC.So what about The Tablet which describes itself as the "International Catholic News Weekly"? As it describes itself as "International" the competant authority to grant permission to use the word "Catholic" is the Holy See (canon 312 CIC). I hope the next addition has the written permission fom the competant authority displayed on the front cover.
Presumably nothing can stop Tina Beattie describing herself as a Professor of Catholic Studies providing she is merely studying Catholicism as a phenonema rather than claiming she is a Catholic Professor or claiming to teach Catholicism. However it does seem strange that having had her invitation to speak Clifton Cathedral cancelled she is now speaking at the invitation of Westminster Diocese's Justice and Peace Commission, presumably, as she has recently become such a controversial figure, with the personal sanction of Archbishop Nichols.
The Motu Proprio does state that the Bishop(s) are responsible for the theological formation and direction of Catholic organisations, they cannot wash their hands of this responsibility and leave it up to ambiguously "Catholic" theological advisors, as Cafod has done and been alllowed to do in the past.