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.
I like to think that a "true" scratch golfer as someone who could go to any course, and play golf at a little above par, or better. Maybe a 3-4 stroke spread one way or the other. Courses they have never seen before, they would consistently play around par.
Yes, that's over simplifying it quite a bit, and I realize slope, and ratings need to be factored in, but scratch is scratch.
There was a guy who could get around the old Winterwood course in Vegas, who could shoot mid to high 60s just about everytime he went out. His card showed him as a scratch player. However, when he played other courses in the area, he was 10-12 shots higher. His scratch scores were on the one the course he was familiar with.