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 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.
As an example without arithmetic progressions, $x^2 + xy + 6 y^2$ and $2x^2 \pm xy + 3 y^2.$ Both represent only primes $23$ and $p$ that satisfy $(p|23) = 1.$ The first represents those for which $z^3 - z + 1$ has a root $\pmod p,$ accounting for $1/6$ of all primes in the long run, the latter pair of forms (both) represent the others, accounting for $1/3$ of all primes. Oh, $23$ itself is represented by the first one.
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.