Belly-dancing or yoga or African friggin' drumming doesn't "belong" to it's culture of origin
Hail lord and mighty Whitey, who can claim anything he wants from any culture he wants. Intellectual property exists only for individuals, after all!
I don't agree with the original article. Like many others, it goes too far. Just about anything in the world can be taken too far, even the very best ideas and the most fundamental right (see: fire, shouting, theater, crowded). This is the same here.
But there is no doubt to me that taking someone else's traditoin, making up your own version without any respect for the actual tradition, and then calling it by the same name as if it were still the same tradition is an immense display of lack of respect for other cultures.
Now, of course, cultures influence each other all the time. Sometime concepts seep from one culture's tradition into another culture's tradition. That's a normal part of interaction between any cultures.
Ultimately, to me, the guiding line should be something like this:
1)If you're going to claim that what you'r edoing or saying actually represent a given cultural tradition: actually represent it.
2)When dealing with the cultures and traditions of smaller groups (such as Native Americans), asking permission and getting the story straight from them is never a bad thing.
3)If what you're doing is just loosely inspired from a given traditional practice, then DON'T use that name for it. Find another name, maybe even a name that allude to the original tradition, but make it clear (somehow) that what you're doing is different, it's own thing, and that you aren't actually following their rules.
4)If your culture absorbed elements of another culture (for example, legends): then it,s fine to use these - they're part of your culture now. but again, it might be wise to hint at the distinction.
For example, in the story I'm working on, I draw a lot on French-Canadian legends. Many of those were influenced by Native names and stories. I intend to note at several points that the creatures that are called "Wendigo" on the basis of how they understood the stories of the Natives; and leave it deliberately hazy whether or not they're actually what the Algonquian people would have described as a Wendigo.