Scratch is 0 handicap. "He plays off scratch" means he plays off a 0 hcp. I don't understand how a scratch golfer could have a hcp above zero. It's more commonly used when describing a competition as not using handicaps.
For all the exacting definitions, people have in their own minds what scratch is, and some of them you can't argue with. And the MSM plays loose with the definition to fit its own narrative, which probably adds to the confusion of casual watchers.