The identity 
$$\log(e^X e^Y) = X + \frac{\text{ad}_X}{1 - e^{-\text{ad}_X}}Y + O(Y^2)\qquad\qquad(*)$$
is due to Poincaré [1], who showed that $W(t)=\log (e^X e^{tY})$ solves the ODE
$$W'(t)=\frac{\text{ad}_{W(t)}}{1-e^{-\text{ad}_{W(t)}}}Y,\;\;W(0)=X.$$
The identity $(*)$ is the solution to first order in $t$. It holds generally, you don't need $[X,Y]=sY$.

[1] H. Poincaré, *Sur les groupes continus*, Trans. Cambridge Phil. Soc., **18**, 220–255 (1900).