> It’s only the former definition that would allow an AI model to have been trained on someone else’s data

There are yet more definitions of "theirs". For example, data whose provenance can be traced back to Anna's Archive.

So the data is legally owned by the book authors, possessed by Anna's Archive, and downloaded for training usage by the AI companies. Every person in that chain could, linguistically speaking, correctly refer to the data as "theirs", or refer to the data of a different entity as "theirs".