I think all cheddars should be cheddared by definition
But I think we can back up and see what the purpose of cheddaring is.
Normally you would take the tomme (ball of curds), stick it in a mold, let it drain while it is knitting and at the same time let the pH drop. You then salt it when it hits the pH you want. I've found that the pH in which you start draining/pressing is cruicial. For reasons that I don't quite understand, when you drain the curd at a higher pH, more calcium is retained in the curds. Even after it has finished draining and you hit your target pH, the total calcium level in the cheese is apparently *higher* than if you started draining at a lower pH. To some extent that makes intuitive sense, but I don't really understand what's happening physically and chemically. Possibly because you create the tomme (ball of curds) early the calcium gets filtered out (and remain in the curds) as the curds are draining. If you leave the curds in the vat, they are all still small and the calcium can leave the curds more easily, but I really don't know.
Anyway, the result (that you can test by trying it many times) is that curds that you drain early will be more pliable and stretchy than curds that you drain late. More calcium remaining means that the when the curd gets down to 5.2 or so, it still has a lot of connection points and so it doesn't get brittle on you. You can especially see that in alpine cheeses. You drain at a very high pH and even though it's a very low moisture cheese, it's almost comically flexible when you get it down near a pH of 5.0.
Enter cheddaring. To my mind (and to be clear, this is my own imagination here), cheddaring is mostly a texture building technique (well Caldwell says the same thing in her book which is what got me thinking along these lines). You make a tomme, then you cut, stack and drain the cheese. Because you have cut the tomme, (and you are keeping the temperature high), the curds drain easily and quickly, but because you have stacked them, I *think* they will probably retain calcium. As you cheddar, you'll see that the curd forms layers. This is (in my mind) assisted by the fact that you aren't pressing it (except under its own weight). When the curd has finished cheddaring, if you tear off a piece it will have striations like a cooked piece of chicken breast. I've posted it before, but Caldwell has a really dramatic picture of a well cheddared slab of curd. It's got 30 or 40 layers in it.
After that you mill the curds. There are 2 different ways of milling the curd. One is to mill it in large pieces. This retains moisture, but also retains that original structure. The second way is to mill it fine. This allows the curd to drain very well, but it looses the original structure. If you do it the first way you will end up with kind of boulders inside the cheese that all have striations oriented in different directions. As the cheese ages, the proteolysis breaks down the striations, but you still end up getting a lot of variation in the "grain" of the cheese. For that reason, when you cut the cheese, it will tend to break apart in large chunks. If you go the second way, you will have a more uniform structure that is grainy and will break apart into small crumbly bits.
If you were to take the same cheese and press it like a tomme, you would get a texture like a tomme. It is practically the same cheese after all, except for the cheddaring! But the thing to consider here is that the tomme will have *much* more moisture because it hasn't drained in the cheddaring process. Both will have a good amount of calcium and have a pliable texture, but the cheddar will break apart, while still being rubbery and pliable (if you know what I mean).
Now imagine a "stirred curd cheddar" (which is not a cheddar in my books, but... oh well). Here you keep the curd in the vat for an extended period and drain the curds extensively in the vat. They are mostly drained of whey before you drain them out of the vat and so you put them in the press immediately, press and then salt. This gives you the same moisture level as a cheddar, but you lack the texture of the cheddar. Importantly, I think you also will lack the calcium content of the cheddar and so you will end up with a more crumbly, less pliable curd.
Now, you can do a kind of hybrid approach, where you intentionally case harden the curds (by raising the temperture quickly), or by doing something like washing the curd. Here you maintain the moisture level of the curd and stop it from draining easily. If you look at a Colby, the traditional technique is to spread out the curds, let them dry out a bit (to further harden the curds and stop them from draining to much) and then press. It's kind of like an anti-cheddar. However, I think it produces a similar result. You retain calcium and you give structure to the curd so that it crumbles while maintaining pliability.
So I think some of the Dubliner approaches are closer to that hybrid approach (if I remember correctly). They want a higher moisture level. They want high calcium content. They want a crumbly curd. It's one of those thins that I think you would really have to watch videos of real production to be able to reason effective about what they are trying to do. It's super subtle. I don't think it's a thing you can reasonably get from a recipe. I feel the same way about good Colby. It's a cheese that's so dependent on technique to get the properties that you are going after.
But... To be fair. You can probably ignore all of that and get very yummy cheddar-like cheese without over thinking :-D. Of course, that's absolutely no fun.
