If $F$ and $F^\prime$ are free $R$-modules, then $M \cong F/H$ and $M \cong F^\prime/H^\prime$ are viewed as equivalent presentations of the $R$-module $M$ if there is an isomorphism $F \to F^\prime$ which carries the submodule $H$ onto $H^\prime$. We study when presentations of modules of projective dimension $1$ over Pr\"ufer domains of finite character are necessarily equivalent.