Equivalence of links in $R^3$ or $S^3$ Definition 1. Two links are equivalent if they are ambient isotopic in $\mathbb{R^3}$. Definition 2. Two links are equivalent if they are ambient isotopic in ${S^3}$. Are these two definitions (essentially) the same?

These definitions are the exact same, because generically an isotopy sweeps a (locally) $2$-dimensional locus, and in a three-dimensional space it can always be made to avoid a particular point. All you have to do is call that point $\infty$.