Press freedom is important because it can keep in check the other branches of power. But it shouldn't be so free as to be able to publish known lies.
A determined litigant can bury you in paperwork, costs, and other hindrances to the point that your speech is essentially surpressed. The OP case is a good example.
Showing up to contest the requested inunction likely cost a lot more than the story is worth. J+J knew they were going to lose this, but they wanted to remind the publisher that they need a lawyer on speed dial. As much as journalists and editors claim they aren't cowed by this kind of behavior, you can bet that they will tread carefully around j+j in the future
I wasn't trying to make a full and ideal rule set, just pointing out that the one you said is extremely narrow and even really bad free speech rules could pass it. Like one that prevents me from talking about someone in a park playing with a kite.
It is absolutely forbidden to report on family court proceedings; there is no public interest defence.
The other category is libel/slander which is properly protected in the UK where a real burden of proof is put on the person making the claim, meanwhile in the USA, "news" shows lie with impunity, and almost always for the benefit of the oligarchy.
Its a better system than the USA's and by far. Right-wing types, who also dominate this forum, dislike it because it limits the lies and racism and islomaphobia they so deeply identify with, and is reinforced to them by the dishonest right-wing media they consume which has almost no limits on dishonest or hateful speech in the USA. So its an ugly cycle of extremists radicalizing new extremists for political gain (read: faithful GOP voters) by the billionaire owners of these media outlets who gain from these narratives both politically and economically.