What I’m speculating on is whether or not the expression “Release a tiger to return to the mountains” was a source of inspiration for the “Embrace tiger, return to mountain” form name. As such, it would be a literary allusion, but that doesn’t necessarily mean that the person who named the movement was highly literate, or familiar with the history book, Records of the Three Kingdoms. The phrase may have had currency as a proverbial saying, or it may have appeared in teahouse stories, street theater, or operas that average people would have had some access to. The Ming historical novel, San Guo Yan Yi, was based on the much earlier history book, San Guo Zhi, and a lot of people were familiar with the characters and themes, so that they become a part of ordinary speech and consciousness. For example, there is a phrase in that novel that goes something like, “Shuo dao Cao Cao, Cao Cao jiu dao.” (As soon as you mention Cao Cao, he shows up.) The phrase “shuo dao Cao Cao” came to mean something like the English phrase, “speak of the devil.” I don’t know where the English idiom “speak of the devil” came from, but I’m familiar with it, and have probably used it. There are idioms in English like, “once bitten, twice shy,” or “a stitch in time saves nine” that people use without thinking about the original sources they came from. They may have proverbial stories behind them, but one can understand the sayings without knowing the source stories.
I’m not aware of a folk story with the theme of returning a tiger to the mountain, but there could be one. There are plenty of references to tigers in mountains of one sort or another. I’m currently in the middle of reading a book by a guy named Anthony L. Schmieg, M.D. titled, _Watching Your Back: Chinese Martial Arts and Traditional Medicine_ (2005, University of Hawaii Press). There is one section in which he discusses the contrast between the traditional Confucian notion of avoiding injury to one’s body, and the impulse for heroic action for a righteous cause. He writes, “The martial hero cries, ‘I know the mountain has tigers, but I must go to Tiger Mountain!’” (Schmieg, p. 26) The author doesn’t say where this came from, but the theme of taking a risk for a righteous cause is echoed in other Chinese proverbs I’ve seen, such as, “You can't catch a cub without going into the tiger's den.” Then there’s “Once on a tiger's back, it is hard to alight.” That one brings to mind the taiji form, “Retreat Astride Tiger.” This seems to acknowledge that certain kinds of actions are inherently dangerous, and that once you engage you must remain committed to the task.
Anyway, if releasing a dangerous enemy (to invite calamity) is a bad thing, “embracing” him, i.e., maintaining control over him would be a good thing. As you say, the scenario fits the application, so the form name makes sense given this context.