I don't think normal is quite enough, but something like that should work. So,
- normal is equivalent to $S_2$ and $R_1$
- reduced is equivalent to $S_1$ and $R_0$
So, if $Y$ is a regular hypersurface in something normal and it is not entirely singular which follows from you genericly reduced assumptiom, then $Y$ is reduced. (This is just restating what you are saying in #3).
So, a simple sufficient condition is to assume that $X$ is at least $S_{t+1}$ where $t=\mathrm{codim}_X Y$. That way $Y$ will be $S_1$ and your generically reduced assumption implies that it is $R_0$.