Great story
The way I've heard that story is they cut off the middle and index finger, and the bowmen would wave both which is what we do in Britain rather than the middle finger
When they caught our bowmen they would cut off the two fingers making the bowman useless, but they would still stand on top of the battlements and pretend to be ready with their bows, the French realised they were being tricked so thought it would be ok to still attack, to prove to the French that they still had their fingers they would wave them at the French soldiers.
We had the best bowmen in the world by a long way, it was the law in Britain that everyman had to practice with the bow everyday
One finger is more American