Yes, I’m also sure that ‘discreet’ would imply keeping mum about info discussed between passengers. In time family chauffeurs were employed less and less by families, along with domestic servants, as services could no longer be afforded. I expect hiring from Hunts when needed would have been the alternative, so customers would want to be assured of the discretion of unknown drivers supplied by Hunts.
So the ‘discreet liveried’ description could imply both discreet in dress as well as in character. Viv.