output impedance without small signal model
Yes - there is another (intuitive) approach based on feedback theory.
As we know - the resistor Rs (I am using a large symbol because it is an ohmic resistive part) provides negative current-controlled voltage feedback. According to feedback theory, the output resistance in this case is always increased by the factor (1-loop gain LG). 

For the given circuit the loop gain is LG=-gmRs and, as a result, we have r,out,d=r,ds(1+gmRs)=r,ds(1+Rs/r,e) with r,e=1/gm. The symbol r,e  is used sometimes instead of 1/gm because 1/gm is the input resistance at the source node for common gate stages.

Similarly, we can find the expression for r,out,s if we replace Rs by Rd and r,e by r,ds. It is a bit uncommon to consider - as far as the output resistance at the source is concerned - the small feedback effect caused by Rd (small loop gain LG=-Rd/r,ds because r,ds is much larger). Nevertheless, that is the way we arrive at the given expression for r,out,s  .