Good example. I see the application of how this is used. I think I am fighting over the language of the 210.19(A)(1) section. To me, when I see the word minimum I think "the least quantity or amount possible, assignable, allowable, or the like," as described by the definition. This means, if you say minimum, then it is acceptable to use at the very least. Part of 210.19(A)(1) says "the minimum branch circuit conductor size, before the application of any ampacity adjustment or correction factors, shall have an allowable ampacity of not less than the noncontinuous load plus 125% of the continuous load." However, this is not entirely true, since of course there is the stipulation of ampacity adjustment after the fact. So while this sentence is used to get a baseline ampacity, it is really is not THE minimum ampacity, as this is finally determined by the ampacity adjustment / correction factors as the final step of this process.
On the contrary, the statement is entirely true. What you seem to still be missing is that the second sentence in 210.19(A)(1) establishes a minimum conductor
SIZE, not a minimum conductor
ampacity.
the minimum branch circuit conductor size, before the application of any ampacity adjustment or correction factors
For example, if you had a 100A load which consisted of 80A non-continuous load and 20A continuous load, and you wanted to use a #3 THHN in a 100degF ambient, the conductor would have a corrected ampacity of 115*0.91 =105A, so #3 THHN has a sufficient ampacity for the load. HOWEVER, the second sentence in 210.19(A)(1) requires a minimum conductor size, before correction, that has an ampacity that is not less than the non-continuous load + 125% of the continuous load. This would be 80A + 1.25*20A = 105A. #3 (at 75deg) has an allowable ampacity of 100A which is less than 105. That would make the minumum conductor size for the circuit #2. You can't use the #3 THHN, even though its corrected ampacity is sufficient for the load, because it is smaller than the minimum allowable conductor size.
The application of the adjustment/correction factors do not change the minimum allowable conductor size.
I have another question.
Check out the example in 220.20(A) after the exception. Check out Step 3 which calculates conductor adjusted ampacity. There's a 25A continuous load in the example, and to calculate the required ampacity of the conductor using the adjustment factor of 80%, it takes the load 25A and divides by 80% to get 31.25A. I don't get this. I thought ampacity adjustment factors are applied to the conductor's rated ampacity based on insulation. For instance, in your example from D3(A) in your post you took 2/0 conductor and applied the ampacity adjustment of .7*.96 to the conductor's rated 90 degree ampacity of 195A, which turned out to be 131A. However, according to this example in 220.20(A), it is taking the load and dividing by the ampacity adjustment.
Multiplying a specific conductor ampacity by correction factors to find the final conductor ampacity, or dividing the required circuit ampacity by the correction factors to find the a required conductor size are just two sides of the same coin.
For instance, in D3(a) the ampacity required for the load is 119A. I could check 1/0 THHN and see that the corrected/adjusted ampacity would be 170A * 0.7 * 0.96 = 114A. That would be too small for the 119A load. Checking 2/0 THHN, I see that the corrected/adjusted ampacity would be 195 * 0.7 * 0.96 = 131A, which is large enough for the load.
The other side of the coin would be to take the required ampacity for the load and divide it by the adjustment/correction factors: 119A / (0.7 * 0.96) = 177A. I would then see what conductor has an ampacity of at least 177A. Looking at 1/0 THHN I see 170A which is smaller than 177, so 1/0 THHN is too small. Looking at 2/0 THHN I see 195A which is larger than 177, so 2/0 THHN is acceptable.
If we did this in your D3(A) example, of the load of 119A / (.7 * .96) = 177A! This would require 3/0 conductor. Something doesn't make sense.
As noted above, the ampacity of 2/0 THHN is 195. So only a 2/0 conductor would be required.