Uhhh, that’s a cleaner. I’ve used it by hand on the other table I had in Lake Placid. The balls were not as slippery as the cleaner/polish I use in the Ballstar. It would be an interesting comparison to make. I would assume there is some polish in what you are using and most likely your home made cleaner does a better job than my hand cleaning did.
Another thing to try is the Ballstar cleaner/polish in your machine....
I'm not a fan of the Ballstar solution. It leaves the balls very slippery. The Tiger product is the best I've found when using a machine.