Personally, only one helmet who bore the name of ''brain bucket'' french name ( cervelière ) was the first French helmet.
It was in the form of bowl attached as head by a cordon.
And it before arriving him from the Adrian in 1915.[You must be registered and logged in to see this image.][You must be registered and logged in to see this image.]
Everything them other helmets have them names and nickname.
As British Mk1 ( Brodie)
US mK1 (doughboy helmet)
Canadian MKII (soup bowl )
The Brodie helmet, called Helmet, steel, Mark I helmet in Britain and the M1917 Helmet in the U.S., is a steel combat helmet designed and patented in 1915 by Englishman John Leopold Brodie. Colloquially, it was called the shrapnel helmet, Tommy helmet, Tin Hat, and in the United States the doughboy helmet. Worn by Australians during WW2 and sometimes known as 'Panic Hat'. It was also known as the dishpan hat, tin pan hat, washbasin, battle bowler (when worn by officers), and Kelly helmet. The US version, the M1917, was copied from the British Mk 1 steel helmet of 1916. The German Army called it the Salatschüssel (salad bowl).
During the WW2 the US M1 helmet was always called as the US M1 or M1 steel pot.