The reason I mentioned the Twitter account was that was how I traced the individuals behind your link, which doesn't give any source.
I am aware that there has been controversy about press regulation since Leveson and the formation of Hacked Off. I also know that some newspapers object. Incidentally, the Times and the Sun are hardly diverse, as they are owned by the same person. News International was forced to pay out large sums to people such as Christopher Jefferies, so of course they object.
It's quite interesting that John Whittingdale is quoted, because pressure was almost certainly used to close down reporting on his own Miss Whiplash story and reporting about his half-brother, a convicted paedophile. 'Ordinary people' don't have that kind of leverage and can't afford to take papers to court. Some posters seem to think that if a paper breaks the law then the 'law' will automatically kick in and prosecute. Of course it doesn't. People need to be able to risk £thousands to take these kind of cases to court and that's what Section 40 is trying to make easier.
I don't believe in unlimited free speech. The right to have a voice comes with responsibilities and I think the media oversteps the mark, which is even more difficult to monitor in the age of the internet. What we're now seeing are 'false news' stories and deliberate hounding of innocent people for sensationalist news headlines.
The people funding the link you posted are self-declared libertarians in a Tea Party/Trump sense. More sinister, however, is the obvious involvement of alt right groups. There is controversy on university campuses at the moment, because certain far right publications have been banned by student unions. They are print copies of publications such as the Daily Bale. The founders are all a bit secretive, but it's widely accepted that the Daily Bale's founder is someone called Joshua Bonehill-Baine. He's currently serving a prison sentence for various crimes, including being behind the anti-semitic hate messages sent to various Jewish MPs, including Luciana Berger. He's published the names of all Jewish MPs and suitable hate messages which could be sent to them. The group has made up all sorts of stories, some of which have found themselves into the msm. The law is very weak on libel and, personally, I think it needs strengthening.
As far as criticism of government and other authorities is concerned, Section 40 strengthens the media's hand, because it limits the payouts. Some argue that it actually empowers them to be more critical. Reading through the msm, there is certainly no shortage of criticism, so they don't seem to have too much to worry about. I'm more concerned about 'ordinary' people and those from minority groups, whom the press harrasses with impunity. Unless some action is taken, we'll end up with a democracy which means the 'tyranny of the majority'.