I think we're nearly all in a similar boat, because traditional classical education did serve such a small minority. By the time any of my known ancestors were able to stay in school past the 8th grade, the system was pretty much defunct everywhere. Even the vestiges, such as Fr. Donnelly's work, had almost disappeared by the 1940s - and the context for understanding them had been completely swept away.
I'm not sure that definitions are what we need, either. The top-down, "theory -> practice" approach might actually be the major problem with modern education. The old-time classical systems weren't based on a definition; they were based on the handing down of specific, embodied practices that had been found effective. In other words, they were more of an art than a science.
Another thing I've picked up from my reading is that all education is local. It took me years to get some sort of grasp of "the history of classical education," and once I had it, I realized that it didn't bring me any closer to answering the question of what to teach my children. To do that, I had to look at the history of specific systems, and then read more detailed descriptions of individual schools within those systems (e.g., Stonyhurst and Boston College for Jesuit education). On what basis were they founded? How were their teachers trained? What books did they use? What was distinctive about their teaching style? What was controversial about them? What was the students' way of life, outside the classroom? What reforms did the school undergo? What were their graduates known for?
At that point, I finally felt able to start making connections with modern books and methods, and with the sorts of things I'm trying to achieve in our homeschool. I've made some gradual changes based on what I've learned, but I think it's important to practice radical acceptance of our lack of understanding, and give up any sense of urgency to "figure it out." Children can do just fine when they're raised and educated without the very best practices in every area. Consistency and calm are worth a great deal. Even if we don't end up helping our children nearly as much as we'd like, we can keep educating ourselves, and inspire them to do the same. And maybe help our grandchildren. Culture is a long-haul project.
Because of all this, it seems to me to be futile for an individual, or even an organization, to try to "restore classical education" in some general sense. We have to choose one or more examples of the tradition - whether or not we have any personal connection to them - and try to renew them, discarding unnecessary additions, restoring things that were lost, and bringing in new ideas from outside. While the "Great Books movement" seems to have been far too big a break (in content, pedagogy, and goals) to count as an example of the tradition, it could be a source of those new ideas.
For the average US homeschooler coming from a Protestant background, the old-time academies might be one interesting starting point. The local academy was the sort of classical school that many board members' ancestors would have attended, if they'd had the opportunity and inclination. Even if they had to work their way through by sawing wood, a la President Garfield.
Yes. I think my problem is insecurity, so that when practices (and results) seem uncertain I grasp more desperately to "figure it out." I do agree though, we have to move forward.