Here are some naive comments. If $X, Y$ are large then the contribution of the quadratic terms swamps the other terms, so first let's concentrate on the quadratic terms $$Q(X, Y) = a X^2 + b XY + c Y^2.$$

This can be written as $$Q(X, Y) = \left[ \begin{array}{cc} X & Y \end{array} \right] \left[ \begin{array}{cc} a & \frac{b}{2} \\\ \frac{b}{2} & c \end{array} \right] \left[ \begin{array}{c} X \\\ Y \end{array} \right]$$

and the matrix occurring above has determinant $ac - \frac{b^2}{4} = - \frac{\Delta}{4}$. $Q$ is positive-definite if and only if $\Delta < 0$, in which case this determinant has concrete geometric significance: it is $\pi$ times the reciprocal of the area of the ellipse $Q(X, Y) \le 1$. Consequently it describes the asymptotics of the number $q_n$ of pairs of integers $(X, Y)$ such that $Q(X, Y) \le n$ (and the same should be true of $F$). In all cases, $\Delta$ is invariant under affine change of coordinates.

$F$ itself can be written as

$$F(X, Y) = \left[ \begin{array}{ccc} X & Y & 1 \end{array} \right] \left[ \begin{array}{ccc} a & \frac{b}{2} & \frac{\alpha}{2} \\\ \frac{b}{2} & c & \frac{\beta}{2} \\\ \frac{\alpha}{2} & \frac{\beta}{2} & \gamma \end{array} \right] \left[ \begin{array}{c} X \\\ Y \\\ 1 \end{array} \right]$$

and $D$ is $4$ times the determinant of this matrix. Again this is invariant under affine change of coordinates. $D$ is a natural invariant of the homogenization of $F$ to a ternary quadratic form.

Here are some naive comments. If $X, Y$ are large then the contribution of the quadratic terms swamps the other terms, so first let's concentrate on the quadratic terms $$Q(X, Y) = a X^2 + b XY + c Y^2.$$

This can be written as $$Q(X, Y) = \left[ \begin{array}{cc} X & Y \end{array} \right] \left[ \begin{array}{cc} a & \frac{b}{2} \\\ \frac{b}{2} & c \end{array} \right] \left[ \begin{array}{c} X \\\ Y \end{array} \right]$$

and the matrix occurring above has determinant $ac - \frac{b^2}{4} = - \frac{\Delta}{4}$. $Q$ is positive-definite if and only if $\Delta < 0$, in which case this determinant has concrete geometric significance: it is $\pi$ times the reciprocal of the area of the ellipse $Q(X, Y) \le 1$. Consequently it describes the asymptotics of the number $q_n$ of pairs of integers $(X, Y)$ such that $Q(X, Y) \le n$ (and the same should be true of $F$). In all cases, $\Delta$ is invariant under affine change of coordinates.

$F$ itself can be written as

$$F(X, Y) = \left[ \begin{array}{ccc} X & Y & 1 \end{array} \right] \left[ \begin{array}{ccc} a & \frac{b}{2} & \frac{\alpha}{2} \\\ \frac{b}{2} & c & \frac{\beta}{2} \\\ \frac{\alpha}{2} & \frac{\beta}{2} & \gamma \end{array} \right] \left[ \begin{array}{c} X \\\ Y \\\ 1 \end{array} \right]$$

and $D$ is $4$ times the determinant of this matrix. Again this is invariant under affine change of coordinates. $D$ is a natural invariant of the homogenization of $F$ to a ternary quadratic form.

Let me also say some naive things about the two conditions appearing in the OP. By the quadratic formula, the quadratic polynomial $ax^2 + bx + c$ has roots $$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$

Consequently, $\Delta$ is a square if and only if this polynomial has rational roots, hence if and only if $Q$ splits into a product of factors $$Q(X, Y) = (dX + eY)(fX + gY)$$

over $\mathbb{Z}$ (by Gauss' lemma). As Will Jagy says, these quadratic forms have very different behavior as far as representing primes compared to quadratic forms that do not factor this way.

If $\Delta$ is not a square, then it is in particular not zero. Now, if $D = 0$, then the $3 \times 3$ matrix above has nontrivial nullspace. I believe this is equivalent to being able to write $F$ as the product of two linear polynomials (after a suitable quadratic extension) as opposed to the product of two linear polynomials plus a constant.