Is the discretization of the Guassian distribution on torus still a discrete Gaussian distribution?
There are two ways to discretize: rounding and conditioning.
The first discretization you use to define $D_{\mathbb Z,s}$ from $\rho_s$ is using conditionning.
The second one defining $\bar \Psi_s$ from $\Psi_s$ is using rounding (the density at integer $x$ of the former distribution is given the total density of the interval $[x-\frac 1 2, x+ \frac 12]$ of the latter distribution).
So, because of this, the two distributions are not the same. However, for large $s$, they would be very close to each other, because they would both be very close to uniform.
To summarize: the moding (ie going from R to the torus) and discretizing do commute, but only if you use the same discretization on both path.