In various places in the literature surrounding the Gross--Zagier formula, the results in Heegner points and the derivatives of $L$-series (hereafter, Heegner points) are referred to as a generalization of the results in On Singular Moduli. Most notably, in Heegner points, Gross and Zagier refer to On Singular Moduli as "a detailed consideration of the $N = 1$ case" of the local height computation in Heegner points.
Looking at the two papers (particularly the algebraic proof in On Singular Moduli and the computation of the local Neron--Tate height at non-archimedean places in Heegner points), I see the similarities in the structure of the proofs, and in the final formulas that derive from quaternionic computations. What I'm missing is the explicit link between the final results of the two papers.
On the one hand, On Singular Moduli computes an exact valuation for the norm of the difference of two $j$-invariants at CM points. On the other, Heegner points computes the local height of certain divisors on the Jacobian of a modular curve. There are several places where my background is less than complete, so perhaps I am just missing some fundamental piece, but I don't see how the valuation computation in On Singular Moduli is the same as the local height computation in Heegner points, except on the curve $X_0(1)$ instead of $X_0(N)$.
Can anyone state explicitly how the result obtained in On Singular Moduli is the $N = 1$ case of the results in Heegner points?
I'll note that there is the hint of an answer in Emerton's response to this question, particularly part (6), but this doesn't quite answer how the local height computation in Heegner points becomes the valuation computation in On Singular Moduli.
