Additive discrete logarithm:
In $\Bbb Z_n^+$ we have to find $z$ in $zg=h\bmod n$ where $g$ generates $\Bbb Z_n^+$. $z$ is unique upto $z \bmod n$.
Multiplicative discrete logarithm:
In a cyclic group $\Bbb Z_n^\times$ we have to find $z$ in $g^z=h\bmod p$ where $g$ generates $\Bbb Z_n^\times$. $z$ is unique upto $z \bmod n-1$.
Elliptic curve discrete logarithm:
In Elliptic curve group of points $E(\Bbb Z_n)$ where $n$ is an integer (may be composite) we have to find $z$ in $zG=H$ where $G$ generates $E(\Bbb Z_n)$. Is $z$ is unique up to $z \bmod n-1$?