Digital BW, The Print

It seems to me that we have usefully isolated two different ways you can
potentially measure the tonal qualities of a print. The first

(1)    D1 = Dmax - Dmin

measures the difference between the brightest and darkest parts of a
print. I think we all have an instinctive feel for what a big number
versus a small number means here. In this context Dmax represents the log
density of the darkest black, Dmin the log density of the whitest white,
presumably the paper base.

The second

(2)    D2 = Cmax - Cmin

where C is a log measure of contrast, measures the difference between the
greatest contrast a printer can render, and the smallest. (In my opinion
this needs to include some criterion of how large an area we are talking
about, in order to take into account the coarseness or otherwise of the
dither, but that's a side issue).

I'm assuming here that 

(3)    Cmax = Dmax - Dmin.

I'm also assuming that 

(4)    Cmin = Dmin' - Dmin

where Dmin' is the lightest tone the printer is capable of rendering
which is distinguishable above the 'noise level' or natural tonal
variation of the paper base.

Plugging (3) and (4) into (2) gives

(5)     D2 = Dmax - Dmin'


Now the question here is not which of these, D1 or D2, is properly called
Dynamic Range, but whether there is a useful difference between them?

For example, what are the perceptual differences between two prints with
different values of D2? Does it really measure tonal smoothness? We need
some concrete examples to be able to decide that I think.