Binary quadratic forms in $\Bbb Z_2$

$\def\Z{{\Bbb Z}}\let\f\frac\let\d\delta$Concerning binary forms, there is something I don’t understand: Let $F$ be a form $F=aX^2+2bXY+cY^2$, $a,b,c\in\Z$ or $\Z_2$.

Let assume $F$ to be primitive, that is $a$ or $c$ odd, and so invertible (or unit) in $\Z_2$. I write $$F=a\left(X+\f baY\right)^2+\left(c-\f{b^2}a\right)Y^2=a\left[\left(X+\f baY\right)^2+d\left(\f Ya\right)^2\right]\sim a(X^2+dY^2)$$ with $d=ac-b^2$, the determinant of the form, or $-D/4$ with $D$ its discriminant.

Two forms $F,F'$ with the same discriminant are equivalent iff their coefficients $a,a'$ differ by a square: $a'=a\tau^2$. Now an integer in $\Z_2$ is a square iff it is a square in $\Z/8\Z$ that is if $a=a'\bmod8$.

But in the two texts I could find about this subject (for instance Burton Jones “Arithmetic Theory of Quadratic Forms”) distinctions are made depending on the values of $d$ modulo 4 or 8 and I don’t understand where is it coming from. Of course, if I ask the question it is because there is no proof: Just “Using theorem X and Y one can easily prove”, followed by a list of the 6 cases of congruence of $d$. Where is my argument wrong ?