I made two corrections in your formula for $D.$ It is now consistent with the 1974 Acta Arithmetica paper.

Some examples. If $\alpha, \beta, \gamma$ are all $0,$ we have a quadratic form, which is required primitive. For positive forms, the proportion of primes represented is a constant times the full count of primes, this is Cebotarev density. See Theorem 9.12 on page 188 of David A. Cox, *Primes of the Form $x^2 + n y^2.$*  

In comparison, $X^2 + Y^2 + 1$ represents a constant times $\frac{x}{( \log x )^{3/2}}$ primes up to $x.$ I was not aware of this. Note that this case was proved by [IWANIEC 1972][1] where the pdf can be downloaded. He does this special case and improves on estimates of Motohashi.  


My understanding is that, for nondegenerate indefinite quadratic forms, that is $aX^2 + b XY + c Y^2$ with $\Delta = b^2 - 4 a c$ nonnegative but not zero or a square, the number of primes $p$ represented and the number of primes $q$ such that $-q$ is represented both obey a Cebotarev-like law. Franz would know details. I am taking primes as positive only. For example, $X^2 - 3 Y^2$ represents all (positive) primes $p \equiv 1 \pmod {12},$ and all $-q$ for positive primes $q \equiv 11 \pmod {12}.$  


If, instead, I take a degenerate indefinite form as $X^2 - Y^2,$ I can represent all multiples of $4$ and all odd numbers. So the number of primes up to some positive bound $x$
is just the usual $\frac{x}{\log x}$ from the Prime Number Theorem, without any constant multiplier. 

Finally, what happens with $X^2 + Y^2 + 2 X + 1$ or $X^2 - Y^2 + 2 X + 1?$ These can be rewritten with $(X+1)^2 = X^2 + 2 X + 1.$

So, overall, they are saying that two distinct cases give PNT times a constant, either degenerate (representing an entire arithmetic progression containing more than one prime) or a genuine nondegenerate quadratic form, which does not represent any arithmetic progression but may represent all primes in an arithmetic progression (such as $X^2 + Y^2$ and $4n+1,$ which behavior requires very small class number) but in any case follows a Cebotarev-like rule for primes.  

The other case is like $X^2 + Y^2 + 1,$ representing fewer primes but still infinite. 

So, they have managed to put together a number of different cases with two constants. 



  [1]: http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-aav21i1p203bwm?q=e38d3128-78b9-4500-a8fc-cae82bd63b64$1&qt=IN_PAGE