I became interested after reading this passage on Wikipedia: "The term neijia and the distinction between internal and external martial arts first appears in Huang Zongxi's 1669 Epitaph for Wang Zhengnan.[12] Stanley Henning proposes that the Epitaph's identification of the internal martial arts with the Taoism indigenous to China and of the external martial arts with the foreign Buddhism of Shaolin—and the Manchu Qing Dynasty to which Huang Zongxi was opposed—was an act of political defiance rather than one of technical classification."
Since the two styles appear to have further diverged and coalesced in the modernization and standardizing of martial arts, were they considered rival styles? I also have in mind the samples used by Wu Tag Clan which includes a kung fu film with the premise based on that exact rivalry.
What does religion have to do with it? Was Buddhism once seen as 'foreign' in China? Is that still the case today?
Oh, man. A question that's like specifically tailored for me. I'm so happy. This is the exact stuff I study. Have I got some stuff to tell you.
So there's a number of things really at question that you've brought up:
The Shaolin/Wudang dichotomy is a classic structural system in the classification system of Chinese martial arts. For centuries, martial arts practices in China were categorized into two different schemas. The first was internal (neijia) and external (weijia). The other was Shaolin and Wudang. These ultimately became tied together as internal practice was related to Wudang and external related to Shaolin. There's was a vast mythology of origin connected to both Shaolin and Wudang in many Chinese martial arts traditions. The two most popular being Zhang Sanfeng from Wudang creating Taijiquan which goes back to what you mentioned in your post and the destruction of the Shaolin Temple narrative which is found in many Southern Chinese martial arts traditions.
However, these traditional mythologies were challenged by Tang Hao in the the 1920's. Through his research, Tao Hong basically exploded the entire histories that had been constructed around the Chinese martial arts and the Shaolin/Wudang mythologies. However, his work and subsequent work by other scholars created such a conflict within the martial arts world that the Chinese government had to step in and say that the issue was no longer allowed to be debated.
However, despite the fact that research by scholars, such as Tang Hao, Stanley Henning (whom you mentioned) and so on, have shown that many of these stories are historically disconnected, it hasn't stopped the proliferation of these mythologies. The major means by which the Shaolin-Wudang dichotomy spread was through the production of wuxia literature (particularly the works of Jin Yong) and movies in the mid-twentieth century.
Now, this doesn't mean there aren't some interesting historical precedents that build into the Shaolin-Wudang rivalry. Buddhism was at one point seen as a foreign religion. During the "medieval" stretch of Chinese history (roughly 250 A.D. to 1368 A.D. depending on who you talk to) there was lots of friction between Daoism and Buddhism. There was a constantly shifting environment of official support of Buddhism and Daoism by various emperors. Some of the major outcomes of the conflict between the two were arguments by Daoists that Buddhism was just a version of Daoism preached by Laotzu when he left China, the Buddho-Daoist debates during the early Tang Dynasty which led to a lot of mixing and adoption of theories between the two, the heavy persecution of Buddhists in the late Tang Dynasty by a couple of emperors, and the adoption of Tibetan Buddhism by the Mongol Yuan Dynasty. Relationships between Buddhists and Daoists were definitely not always peaceful and it's quite plausible that this history fed into the Shaolin-Wudang rivalry that we see in later wuxia literature and film. When it came to the beginnings of the Manchu-run Qing Dynasty, there is some information to suggest that the Manchu did have concerns over centers of religion like Shaolin and Wudang becoming hotbeds of revolutionary fervor because religion had often been the catalyst for many political rebellions throughout Chinese history.
These days, there's still something of a perceived rivalry between the two, though not many people are really getting into fights over it. People practice Buddhism and Daoism side-by-side as part of everyday practice. Both Shaolin and Wudang have been adopted by the Chinese government tools in the economy and culture. Images and ideas of both temples have been exported around the world and both temples have even produced series of instructional videos that you can purchase!
There's still much that escapes our historical microscope when analyzing the history of Chinese martial arts, but there's still a lot of fascinating information out there. I hope this has been helpful!
Suggest Reading:
Anything by Stanley Henning
Judkins, Benjamin N. and Jon Nielsen. The Creation of Wing Chun: A Social History of Chinese Martial Arts. Albany: State University of New York Press, 2015.
Kang, Gewu. The Spring and Autumn of Chinese Martial Arts. Santa Cruz: Plum Publishing, 1995.
Lorge, Peter Allan. Chinese Martial Arts: From Antiquity to the Twenty-First Century. Cambridge: Cambridge University Press, 2012.
Shahar, Meir. The Shaolin Temple: History, Religion and the Chinese Martial Arts. Honolulu: University of Hawai’i Press, 2008.
Anything by Douglas Wile