= Solution
Let $f$ be the nonzero bounded <holomorphic function> in the zero-set condition. Choose $v$ with $f(v)\ne0$ and compose with a disk <Möbius transformation> carrying zero to $v$. Call the resulting function $g$ and its zeros $a_n$, counted with multiplicity. Then $g(0)\ne0$ and $|g|\leq1$.
For a radius $r$ with no zero on its circle, <Jensen's formula> gives
$$
\sum_{|a_n|<r}\log\frac r{|a_n|}=\frac1{2\pi}\int_0^{2\pi}\log|g(re^{i\theta})|\,d\theta-\log|g(0)|\leq-\log|g(0)|.
$$
To see the formula directly, divide $g$ by its finitely many interior zero factors. The logarithm of the modulus of the resulting zero-free <holomorphic function> is harmonic near the closed disk and satisfies the <mean value property>. The circle average of $\log|re^{i\theta}-a|$ for $|a|<r$ is $\log r$, obtained by expanding $\log(1-(a/r)e^{-i\theta})$. Subtracting its value at zero gives exactly the displayed zero sum.
Let $r\uparrow1$ through zero-free circles. Increasing limits of the nonnegative summands give
$$
\sum_n-\log|a_n|\leq-\log|g(0)|<\infty.
$$
Since $1-|a_n|\leq-\log|a_n|$, the <Blaschke condition> follows for the transformed sequence. The <hyperbolic metric> identity and the base-point comparison proved above give
$$
\boxed{\sum_n e^{-\rho(w,z_n)}<\infty\quad\text{for every }w\in\mathbb D.}
$$
This proves the reverse implication and completes the equivalence.
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