In 1989, Manin and his collaborators formed a series of conjectures on the asymptotic behaviour of the number of solutions to diophantine equations. Let $X\subset \mathbb{P}^n$ is a fano variety (that is $-K_X$ is ample) under its anticanonical embedding, and let $H$ be the associated height function. Then it is expected that there exists a zariski open subset $U \subset X$ such that the number of rational points of height less than $B$ is asymptotic to $c_X B(\log B)^{r_X},$ as $B \to \infty$ for some constants $c_X$ and $r_X$.
This result is true in some cases, for example complete intersections with many variables using the circle method, and also for some del Pezzo surfaces. But there are counter-examples showing that it is not true for all fano varieties.
At any rate, Peyre formed a conjecture on the leading constant $c_X$ which occurs in the asymptotic formula. It is very close to what you describe. One defines a measure on the set of adelic point on $X$, and then the leading constant contains the volume of the closure of $X(\mathbb{Q})$ inside the adeles.
However, there are also some extra factors $\alpha$ and $\beta$ present, related to the position of the anticanonical divisor in the effective cone and the Brauer group of $X$. For conics with a rational point, we have $\alpha=1/2$ and $\beta =1$. Hopefully this should explain your missing factor of two.
Papers:
J. Franke, Y. I. Manin and Y. Tschinkel, Rational Points of Bounded Height on Fano Varieties. Invent. Math. 95, 421--435 (1989).
E. Peyre, Hauteurs et measures de Tamagawa sur les variétiés de Fano. Duke Math. J., 79(1), 101--218 (1995).