Double Think about Numerosity
If I understood the question correctly, the OP is asking for one of these:
- An established name for the "Inverse function rule" stated in the question
- An established name for a concept of "numerosity" based on that rule
- A resolution to the discrepancy between this concept and set theoretical cardinality.
1.
The expression $\lim_{n\rightarrow\infty}\tfrac{D(n)}{A^{-1}(n)}$ raises the question of how is $A^{-1}(n)$ defined for $n$ not in the image of $A$. The theorem seems to state that any extension of $A^{-1}$ to a monotone function on the reals is asymptotically equivalent to $D$. I am inclined to believe there's no established name for this result in the literature.
2.
I don't believe there's an established name for this either. While similar to natural density, the "numerosity" here is given not as a number, but as a function (or rather, an asymptotic equivalence class). There exist other examples of classifying objects by asymptotic behaviour, computational complexity being the main exponent.
3.
The answer would depend on what question is one trying to address by the concept.
Cardinality is the natural answer to the question "how many elements are there in a collection?", based on the idea that if you can put the elements of one collection in one-to-one correspondence with the elements of the other, the collections are the same size (the OP's own diagrams illustrate this quite well). This makes no allusion whatsoever to any properties of the elements, or their relation to one another—it only requires that enough elements exist on each side to match the other side. One can rearrange or relabel the elements of a set at will, and the cardinality will remain unchanged. Much of set theory is about the question of when and how can one construct such one-to-one correspondences, to the point where one can argue that the only really "interesting" property of a set is its cardinality (more precisely, the category of sets is equivalent to the category of cardinal numbers).
Both natural density and "numerosity", on the other hand, are different properties of the distribution of a subset of naturals within the whole, namely different ways to answer the question "how often does one encounter elements of this subset among the naturals?", likely to be of interest in probabilistic number theory. Both depend strongly on the order structure of the naturals—if we were to rearrange the naturals (and the elements of $A$ accordingly), both measures could vary wildly. Each measure corresponds to a different form of approximation of the relative frequency of finite prefixes of the naturals, one as a single number and the other as a curve. In the latter case, the fact that the distribution of a subset generated by a monotone function $A$ is approximated by the inverse function $A^{-1}$ seems akin to the relation between inverse functions and their derivatives.
There are some stabs being made at expanding the idea of asymptotic densities and measure to account for our basic intuition that the whole should be greater than the part, even for infinite sets. Vieri Benci, Mauro DiNasso and Marco Forti, among others, are doing work along these lines.
A rather advanced treatment is here http://arxiv.org/abs/1011.2089
Recently, this nice, readable article made its way into press at the Journal of Logic and Analysis: Elementary Numerosity and Measure http://arxiv.org/pdf/1212.6201.pdf