When I first came across with the term Circle of Confusion in 2008, I did a great deal of reading about it as well as DoF. And all the articles, opinions from other sites make me believe that is the correct.Who am I to shatter your strong beliefs, Dao, when even the visual examples weren't able to do that ?
You perfectly understand the sensor to print process and how a CoC is determined, but you don't get the optics ! After the CoC is established, the lens is built to reach that value. CoC is a characteristic of the lens. It represents the maximum spot diameter of the projected image of a focused point onto the surface of the film/sensor. It's actually the diameter of the spot of light. And it DOES affect the DoF !!! I will give the examples at the end of this post.
Once and for all, the Circle of Confusion is a LENS parameter. When a lens is built, it's built for a specific camera format. It's calculated so for a fixed size photo it produces the same clarity. So, if it's built for a FF camera the CoF is ~0.03 and if it's built for a APS-C camera the CoF is ~0.02.
You are talking about DoF from the perspective of the final print but take into consideration that a lot of cropping can be done in between taking the photo and printing. You cannot say that for every different crop you may have a different DoF. If I print the two photos from the previous post at a 4x6 format they will look the same. They will have the same depth of field.
Let me give you another example. The DX crop of a D3x 24.5MP photo is a 10.5MP photo and a D80 photo has 10.1MP. The D3x DX cropped photo and D80 photo would have the same size and roughly the same pixel density, therefore you can print them at the same paper size. Where is the difference now ?
Now let's get back to the last part of your statement: "Lenses are indeed designed for optimum performance and coverage for a particular format, but this has more effect on their maximum resolution and may have no effect on DoF, simply because DoF is all about objects that are not in perfect focus."
Now let me show a couple of pictures and then you tell me if the different DoF has anything to do with the different lenses:
Same camera Nikon D80, same distance (not measured but I didn't move the tripod nor changed camera angle or elevation), same focal length 135mm, same aperture f/8, same photographer ().
No wonder they call it circle of "confusion". I am confused as heck!