Technology

Though the headnotes were drawn directly from uncopyrightable judicial opinions, the court analogized them to the choices made by a sculptor in selecting what to remove from a slab of marble. Thus, even though the words or phrases used in the headnotes might be found in the underlying opinions, Thompson Reuters’ selection of which words and phrases to use was entitled to copyright protection. Interestingly, the court stated that “even a headnote taken verbatim from an opinion is a carefully chosen fraction of the whole,” which “expresses the editor’s idea about what the important point of law from the opinion is.” According to the court, that is enough of a “creative spark” to be copyrightable. In other words, even if a work is selected entirely from the public domain, the simple act of selection is enough to give rise to copyright protection.

The court distinguished cases holding that intermediate copying of computer source code was fair use, reasoning that those courts held that the intermediate copying was necessary to “reverse engineer access to the unprotected functional elements within a program.” Here, copying Thompson Reuters’ protected expression was not needed to gain access to underlying ideas.

https://natlawreview.com/article/court-training-ai-model-based-copyrighted-data-not-fair-use-matter-law

It sounds like the case you mentioned had a government entity doing the annotation, which makes it public even though it's not literally the law.
Reuters seems to have argued that while the law and cases are public, their tagging, summarization and keyword highlighting is editorial.
The judge agreed and highlighted that since westlaw isn't required to view the documents that everyone is entitled to see, training using their copy, including the headers, isn't justified.

It's much like how a set of stories being in the public domain means you can copy each of them, but my collection of those stories has curation that makes it so you can't copy my collection as a whole, assuming my work curating the collection was in some way creative and not just "alphabetical order".

Another major point of the ruling seems to rely on the company aiming to directly compete with Reuters, which undermines the fair use argument.