I don't doubt Iommi's entitlement to the name, E-Z, but with Dio, they became a different band. In his biography, Iommi says that several albums began as solo work, but were released by the record company under the Sabbath name. I suppose the band name was also signed to the company and was part of Iommi's contract. He seems to have had a very poor business sense and been a poor judge of character regarding managers.
In at least one of the Ozzy Osbourne biographies or in Iommi's Iron Man book (I cannot remember which, it could be both), we are told that Osbourne gave ideas for songs to Geezer Butler and also sang scat for the latter to turn into lyrics. Tony Iommi, in his biography, reveals that he often worked on material with Gerald Woodroffe, but whether this included lyrics I am not sure.
I do not know who came up with the lyrics on the Osbourne solo albums as I do not know much about them. Bob Daisley wrote a lot of them, and Lemmy contributed some credited with others uncredited(?). With the final passing of the Daisley era, there seems to have been a revolving door of 'writers'. Bob Daisley has an autobiography and I would love to get my hands on a copy.