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The paper "Some heuristics on elliptic curves" (sec 4), has a discussion of ordering by discriminant versus conductor, and the author concludes they should all be the same for the therein purposes (rank 2 curves). In 3.4 (second paragraph), the naive height and discriminant distinction is glossed barely. The paper does not describe Faltings height explicitly, though a real period (not volume) occurs in the analysis of rank estimates and $L$-function vanishings. There is an arXiv preprint, but it lacks some later revisions, I detect. magma.maths.usyd.edu.au/~watkins/papers/heur.pdf
I don't think the integral you have written down has very proper convergence properties, on the vertical lines. If you smooth the counting function, with a Mellin transform that decays enough as $t\rightarrow\infty$, then I think I agree that the zeros are more closely linked.
