Why is it $v_{\rm in} = v_{gs} + g_m v_{gs} R_s$ and not $v_{\rm in} = v_{gs}-g_m v_{\rm gs} R_s$ for the common source amplifier?

It's the relative signs that matter here. For $v_{\rm in}$ positive, $G$ is more positive than $S$. Take $i_1= g_m v_{gs}$ as the current circulating counterclockwise in the right-hand loop (up through $R_L'$, left across the current-source, down through $R_s$ to ground).

If you start with $v_{\rm in}$ on the left-hand side, a Kirchoff loop gives $v_{\rm in} - v_{gs} - i_1 R_s = 0$, so $v_{\rm in} = v_{gs}+ g_m v_{gs} R_S$.