The BBC does have freedom from persecution or prosecution (assuming it doesn't publish things that are e.g libelous). But, due to the particular way it is funded, it doesn't have freedom from government intervention.
In particular the BBC charter is reviewed every so often. This process involves the government and can affect things like the level of funding available to the BBC, and its remit in terms of the services it is allowed to offer. It's no secret that the current government are not big fans of the BBC, both for ideological reasons, and because of cozy relationships with certain commercial broadcasters. They are already waging a war in the press trying to convince the public that the BBC has an intrinsic left-wing bias, and are mooting ideas like preventing it from making programmes that are too popular and so may draw audiences away from commercial rivals. In this environment, and with funding already being cut, and jobs lost, it's not hard to see why you would think twice about running stories that could lead you open to allegations of an anti-government agenda.
Although the particular details of that situation are unique to the BBC, it's important to note that commercial broadcasters, and media outlets in general, also don't have freedom in the sense of freedom-from-consequences. For example there was a story last year about HSBC pulling adverts from media outlets which ran articles critical of its business practices. In an environment where, particularly for print media, margins are tight or non-existent, it's easy to see how this can give commercial interests hidden editorial control over a "free" press.
Not really, freedom of press is much narrower. What you're suggesting is an ideal world where journalists are protected from the egos and reputations of powerful people who might want to retaliate against them for their reporting. There are many forms of legal retaliation that freedom of press cannot protect against. In practice, this freedom is weighed against the consequences of publishing any single piece.
For example, not having to run all (or huge parts of) your stories through a censor before being allowed to publish them (e.g. China, Israel).
Or, like in Germany, various protections like e.g. the police not allowed without complex judiciary checks to monitor the phones and internet connections of a journalist, or to go into a newspaper office and seizing random data to uncover a source.
You can be prosecuted for reasons entirely unrelated to your publishing of a story, even if that is the true reason. Do you think people don't try to intimidate journalists?
