Distribution of zeros of entire and holomorphic functions on the unit disk with restrictions on their growth from above

Let $M$ be a function on a domain $D$ of the complex plane $\mathbb{C}$ with values in the extension $\overline{\mathbb{R}}:=\mathbb{R}\cup\{\pm\infty\}$ of the real axis $\mathbb{R}$, and let $Z$ be a distribution of points on $D$. Two similar but different problems will be discussed — to describe the conditions for the existence of a nonzero function $f$ holomorphic on $D$ with the bound $|f|\le\exp M$ and with the property: 1) the distribution of zeros of the function $f$ coincides with $Z$ or 2) the function $f$ vanishes on $Z$ (all taking into account the multiplicities of zeros). First, our criteria will be formulated for these problems. Approaches to the proof of these criteria also use the apparatus of abstract potential theory (balayage$=$sweeping out, duality, etc.) by some analogy with probability-theoretic methods. Next, we will consider some aspects of these criteria for cases where the domain $D$ is the plane $\mathbb{C}$ or the open unit disk $\mathbb{D}$ in $\mathbb{C}$. The main goals of this adaptation are to achieve clarity and geometrization of the criteria, taking into account the invariance of the plane $\mathbb{C}$ and the disk $\mathbb{D}$ with respect to rotation centered at the origin.