Hier die allgemeine Definition für "Presserecht" durch ein Anwaltsbüro (zweisprachig umschaltbar):

Press Law < Intellectual Property - FPS. Legal advice. Made for you. (fps-law.de)

Siehe auch

Press Law (Concise College Texts) Paperback 

Press Law (Concise College Texts): Amazon.co.uk: Smith, Robin Callendar: 9780421234505: Books

und vor allem

Leveson report calls for new press law

Leveson report calls for new press law | Leveson inquiry | The Guardian

Im konkreten Fall ziehe ich aber die 2. Variante des Vorschlags von mad vor.

Aus dem ersten Link in #2 :

https://fps-law.de/en/practice-areas/intellec...

 Press Law

 ... We counsel clients on how to create media content that avoids infringements of third-party rights, and we represent clients in "right to respond" cases, actions for injunctive relief and correction, as well as lawsuits seeking to recover damages for infringements of privacy and publicity rights. In-depth knowledge of the media industry allows FPS to expertly draft media-related agreements, e.g., cross-media cooperation agreements and contracts with freelance authors. ...

... und der dort verlinkten deutschsprachigen Version :

https://fps-law.de/de/kompetenzfelder/intelle...

 Presserecht

 ...  Hierzu gehört die Beratung bei der Erstellung medialer Inhalte sowie die Prozessvertretung in Gegendarstellungs-, Unterlassungs- und Richtigstellungsverfahren und in Schmerzensgeldprozessen wegen Verletzung des Allgemeinen Persönlichkeitsrechts. Besonders bei medientypischen Verträgen z.B. zu Cross-Media-Kooperationen oder mit freien Autoren, bieten wir profunde Branchenkenntnis und eine interessengerechte Beratung. ...

The complainant had successfully brought defamation proceedings against the newspaper in relation to an article published in 2008. The High Court established that the defamatory meaning of the article was that there were reasonable grounds to suspect...

https://www.ipso.co.uk/rulings-and-resolution...

Sarah Palin's libel case against The New York Times will be dismissed after the jury determines a verdict on Monday afternoon

https://www.dailymail.co.uk/news/article-1051...

Have you been defamed by the press? | Brett Wilson LLP

Our specialist defamation solicitors provide expert legal advice and representation for individuals who have been defamed in the press.

https://www.brettwilson.co.uk/services/defama...

Defamation, Slander & Libel on Social Media - Saunders Law

In such cases, a defamation claim may be the best course of action to obtain a legal declaration that comments made on social media are false and to prevent ...

https://www.saunders.co.uk/services/media-law...

The terms libel, slander, and defamation are frequently confused with each other. They are all similar in that they all fall into the same general area of law that concerns false statements which harm a person’s reputation. This general area of law is called defamation law.

https://www.thebusinesslitigators.com/libel-v...

In England und Wales sowie teilweise in Schottland gilt (seit 2014) der "Defamation Act 2013".

https://www.legislation.gov.uk/ukpga/2013/26/...

"Defamation (libel; slander)" kann (ebenso wie "trespass") auch eine Straftat sein.

Leicht OT, aber vielleicht von Interesse für alle meine geschätzten Vorredner:

Eines der beiden Urteile in einem zivilrechtlichen defamation-Fall:

Wright v McCormack Final Judgment for Hand Down (judiciary.uk)

Ordinarily, a claimant in Dr Wright’s position would be entitled to substantial damages. In this case, however, I decided that he should have only nominal damages of £1. The reason was that, in an attempt to establish that Mr McCormack’s publications had caused serious harm to his reputation, an essential element of a defamation claim, Dr Wright had advanced a deliberately false case until shortly before trial.

Die im OP erwähnte Zivilklage verlangt die Unterlassung von bestimmten Behauptungen (also wohl auch ein Fall , der unter defamation fällt).