I'm just speculating here, but in the case of the low-density home care as you've described it, I suspect the owners prevailed against the HOA because the arrangement in question doesn't qualify is a "home-based business" by any objective measure. I'm actually surprised that the HOA even pushed to enforce that provision in an arrangement like that, because it sounds like the HOA rules as they were written could technically prohibit people from even hiring babysitters for their own children.
A babysitter is an outside contractor who comes in your home to provide service, like a plumber. A home-based day care takes money at that address for services provided to someone else who drives to that address and receives services at that address.