If one is saying "proper action", it would indeed be surprising to require the map to be closed.
However, in the application of this property that I know, what matters is getting a Hausdorff orbit space, which comes from $\{(gm,m)\,|\,g\in G, m\in M\}$ being a closed subset of $M\times M$.
If $M$ is sequential, so is the product $M\times M$ and any proper map into it will be closed. In this case "proper" seems to be stronger than needed.
Conversely, if $M$ is not sequential, $\{(gm,m)\,|\,g\in G, m\in M\}$ may not be closed even if the map $(g,m)\mapsto (gm,m)$ is proper, so the orbit space will not be Hausdorff.