I ask this because a book on Legal English says "claim" is the modern equivalent for "action", but doesn't explain it. If the book's right, then I could argue that "file a claim" means exactly the same as "bring an action".
Page 69, Legal English: How to Understand and Master the Language of Law
I think I'd better draw some examples from COCA. To me, filed a claim doesn't sound equivalent to "bring an action":
Halverson's bailiff, a black man, says he was ordered to rub her feet, give her back massages, put on her shoes, change her oxygen bottles and pick up papers, cookie crumbs and sunflower seed hulls strewn on the floor of her chambers. He eventually filed a claim for discrimination based on race, religion and sex.
She sued this supermarket, claiming her foot had been run over by a cart. And at the local bank, she says her foot was hit by this door, so she sued about that, too. She filed a claim against the city of San Francisco, saying she slipped in a puddle at City Hall, spraining her ankle and tearing her nylons.
The family attorney says he will likely file a claim in a matter of weeks alleging the federal government is liable in Terry's murder because gun found at the scene were part of "Operation Fast and Furious."
The McCoys, along with two other Chicago-area Wave investors, say they plan to file a claim against Web Street for arbitration by the National Association of Securities Dealers. They contend that Web Street misinformed them about the transfer process.