Other than 

So, what you have is that 

 - *normal* is equivalent to $S_2$ and $R_1$
 - *reduced* is equivalent to $S_1$ and $R_0$


At the end the point is that you have to ensure that $Y$ is not entirely singular and I don't see a good way doing this other than assuming that it is cut out by general hyperplane sections. Once you have that, then you need that $X$ is at least $S_{t+1}$ where $t=\mathrm{codim}_X Y$.