Only it is not based on touch. The "Reiki Master" quoted in the article says it works just as well at a distance. He also claims that it works on animals and plants and that many cats can perform Reiki on their owners.
I'd also add that we don't have to let them determine our terminology, either. If the touch parts of Reiki work, then there's no reason we should add an air of validity to all of Reiki by saying "Reiki works" instead of "touch therapy works" or something similar.
Our cat incessantly "kneads" people (moves her feet up and down on us) and I am now thrilled at the idea of charging people for Cat Reiki
Cats cuddle with you. They enjoy being petted. So do dogs. And so do humans! I have a dog whose primary pleasure in life is snuggling up to you on the couch and getting petted. This is not surprising; dogs, like humans, are social animals. And this is a symbiotic relationship. It's pretty well known that therapy dogs significantly reduce stress, including physical markers of stress such as blood pressure and cortisol levels.
Hugs are also beneficial and cause documented positive chemical changes in our bodies. No imaginary mystical forces or increased "vibrations" needed.
It's just a different theoretical framework for describing an observed phenomenon. Even if it's not technically correct, it can still be a good enough cognitive heuristic to help people intuit what's happening or how to influence it.
Think about it kind of like Newtonian mechanics, which are also not quite right but make predictions that are generally close enough for most of what we do in everyday life and much easier for people to work with intuitively.
Granted, most of it IS fake quackery and superstition, and whatever the thing is the pharmacological intervention is probably going to be more effective in a more predictable and repeatable way than most traditional practices. But there's clearly some measure of relief people get from a lot of the traditional systems that we could learn from, even if it means we do a little dance and chant while we administer the ibuprofen.
Cats don't have owners; they have servants.