I'm confused; I thought the BBC was the public channel and Channel 4 (formerly ITV) was the private one, and that's what made them distinct back in your equivalent of our "only three channels" era. Am I to believe now that Channel 4 is more like your version of PBS?
BBC and Channel 4 are both public. The difference is that the BBC is funded by the TV licence (and thus airs no advertising), and Channel 4 is funded by on-air advertising.
ITV isn’t connected to Channel 4.
If you want the history, basically… We had three TV channels until Channel 4 launched in 1982, BBC1, BBC2 and ITV, but the latter was never called ITV onscreen (it solely used the name of the regional franchise, eg Thames for Greater London, Granada for northwest England, etc, until 1999-2001). Channel 4 has always been public, but was set up originally to be more ‘intellectual’ channel, and air programmes that BBC and ITV wouldn’t commission or broadcast. It’s become much more populist since the mid ‘90s, however. But to its credit, it still makes stuff that is unique, and it’s new programming is the only one left on the mainstream channels still aimed at adults.