Peking duck and Pekin duck are only confused in English. In fact, they were originally both the same till some not-so-bright spark came up with Pekin for the breed in order to reduce the confusion. One of modern life's great success stories. Not.
Even worse is that neither the dish nor the breed originated in Beijing and certainly not anywhere called Peking. I wish I had a yuan for everytime I've been asked why and when the capital of China changed its name. It didn't. It has always been pronounced Beijing - all that changed was the transliteration system used in the west from the unreliable Wade-Giles to to the more accurate Pinyin. Peking was never officially accepted as the name, although Peking University is allowed to retain that name for use in English, due to its international fame. It is one of a handful of names not switched to Pinyin. Tsingtao Beer is actually Qingdao beer in China and Moutai, the strong spirit is Maotai in Pinyin.
The duck breed's exact origin is uncertain but almost certainly somewhere in south-east Asia (perhaps China, but probably not), far from Beijing. It came to the atttention of the imperial court and slowly gained in popluarity.
The dish, Beijing duck originated in Nanjing, the Ming Dynasty capital, then migrated to Beijing when the emperors relocated, establishing Beijing as the new capital. In Beijing, the dish became more available to those outside the court and rapidly gained popularity. It was then it was given the Beijing name.
Regarding the lack of confusion in Chinese, the duck is called 北京鸭 (běi jīng yā), literally 'Beijing Duck' in Mandarin while the dish is always referred to as 北京烤鸭 (běi jīng kǎo yā ), literally Beijing Roast Duck.
北京烤鸭 ("Peking Duck") in a Chinese Walmart store
Some Chinese ducks (10 hens and two drakes) were imported to the USA in 1872 and a breeding program put in place. This mostly took place on Long Island and gave rise to the American Pekin duck. At the same time ducks were imported to the UK. Some of the UK ducks found their way to Germany where they were bred. Today, nearly all Peking ducks in Europe are German Pekin. They have even been reimported to the UK. Chinese, American and German Pekins vary due to the different breeding programs utilised.
The Chinese Pekins, as still used in Beijing's restaurants are, of course, the Chinese variety and are less fatty than the US and German types.
Chinese Pekin Duck Legs
My local shopkeeper with two live Pekin Ducks.