I don't know if you're wrong or I'm wrong but a Hackney Carriage usually only refers to the black cab style taxis, and I've never known anyone drive one of those as their private vehicle?
For regular cars that are used as private hire/taxis, the companies where I live have the drivers driving a company-owned car, not their personal car, though I guess some taxi companies do have drivers driving their own cars, especially smaller firms. But those ones wouldn't be called Hackney Carriages.
No idea to be honest. Reading the rule book I posted yesterday (dated 2011 incidentally) a number of rules don't seem to apply anymore. It says Hackney cabs have to be all/mostly black but look at the below taxi rank on Google. I believe I'm correct only Hackney cabs can use the ranks?