\DeclareMathOperator vs. \operatorname*

Question

To define my own math operators, I usually use amsmath's \DeclareMathOperator. However, with \DeclareMathOperator{\Res}{Res} does not give me the desired "limits" under the operator even in displayed math.

Searching this site, I found \operatorname* (and \operatornamewithlimits) which implements the desired behavior.

\begin{equation}
    \operatorname*{Res}_{x = y} \frac{1}{x - y}
    = 1.
\end{equation}

gives

However, I'm still unsure if this can or should somehow be combined with \DeclareMathOperator and generally when to choose \operatorname* over \DeclareMathOperator (besides this particular case).