The bird is the same in both languages. In UK they are properly tagged as bantams from the city in China they came from.Namely Peking forbidden city. The confusion arising from American influence. The Cochin hails from India is three times the size of our Peking bantam and only the fact that both birds have feathered feet that the mistake was made.Someone thought that a bantam breed had originated from Cochin via the larger bird.
No way did they bother to check the already registered Peking Bantam that arrived direct from Peking Palace via British Army during the Opium wars. Presented to Queen Victoria she kept six with cock bird at Osborne House. The French also took a few from the Palace but confusion arrived when Yank entered his Peking bantam, from UK of course, as Cochin small in poultry show.
Now that echoed around and came to the ears of a European breeder who not surprising called them by the all powerful American name of small Cochin. Hope that helps you see that any big mistakes come from those who think they have all the answers to everything .God help the rest of us.