Canyoneering—in the US—and canyoning in the rest of the English-speaking world is an outdoors activity in which participants climb, hike, swim and otherwise navigate canyons.
As noted in the Wikipedia article, this activity is used most often to refer to technical (i.e., difficult and requiring special skills) activities within a canyon such as rappelling (US)—abseiling (UK)—and technical jumps.
The Japanese term キャニオニング, however, does not seem to fit this definition exactly. It is a river sport according to the Japanese Wikipedia article: “キャニオニング（Canyoning）とは、フランス発祥のキャニオン（渓谷）で遊ぶリバー・スポーツである。”
Indeed, if you look at Google Images for キャニオニング, what you mainly see is people sliding down streams and rivers, sometimes alone and other times in human trains. Here are some examples:
In contrast, such images show up hardly at all if you search on canyoneering. Instead, you see images of people rappelling down cliff faces or waterfalls in canyons.
But search for canyoning (again, the non-US version of canyoneering), and you see a mix of the two.
Like so many words from English, it seems that キャニオニング has taken on a character distinctly Japanese, something like a fuzzy faux ami. The Wikipedia article does note that the sport originated in France, and searching for images of canyonisme reveals a mix of photos similar to the mix found with (non-US) canyoning.
Perhaps it is best to keep in mind terms such as waterfall sliding and river sliding. While not as common as canyoneering/canyoning, they may be more accurate in many cases.
Japanese-English Translation Blog 
I wonder if this term might be on par with ナシ versus pear, in which case the English term for the Japanese fruit is distinguished as “Japanese pear”; similarly, a futon in The US often includes a wooden frame to support the cotton pad, while in Japan it is just the cotton matting. I suppose we could call it a “US futon” or something similar, and similarly Japanese canyoning would be what it is, while canyoneering is what it is, and canyoning is again a different flavor.
Thank you for the thought-provoking question!
The Wikipedia article on false friends talks about semantic change and narrowing. One example is the German Hund ‘dog’ and the English hound ‘hunting dog.’ It says that the English member of this pair has a narrowed meaning.
In the 布団-futon pair, English has widened the semantic scope of the Japanese. (Confusingly, a futon bed frame in Japan can be called a ベッド as seen at CAUインテリア.)
For ナシ-Japanese pear, there is a one-to-one relationship as you would not refer to a Japanese pear as a pear. The word retronym describes cases such as acoustic guitar, where guitar once meant ‘acoustic guitar’ but is now modified to differentiate it from electric guitar. There does not seem to be a similar word for cases like ナシ-Japanese pear, but it would be useful!