Partially reprinted from my comment on www.best-sellingbooks.com/the-girl-who-played-with-fire-review.html/comment-page-1#comment-8. Click on the cover images to enlarge them.
The title of the first book was "Men Who Hate Women."
The second one was indeed called "The Girl Who Played with Fire" when published in Sweden, although Stieg's original title was "The Girl Who Fantasized About a Gasoline Can and a Match."
Note the very modern-style dragon, much more to Lisbeth's taste than the Chinese one on the UK cover, I think.
Stieg's original title for the manuscript of #3 was roughly "The Queen in the Air Castle," which ended up being published as "The Air Castle That Was Blown Up" -- neither one really suitable for use in English.
So you can see that "The Girl..." was used originally only on the 2nd book. The titles of the French editions seem to have been taken from earlier versions of the manuscripts rather than the later books as published in Sweden, since I believe France was the first country outside Scandinavia to translate the trilogy. There will be plenty of work for literary scholars of the future to unravel the differences in all the versions and translations!
Black Mirror: Crocodile (Netflix)
37 minutes ago