Probability distribution function in Regev Cryptosystem
Neither of the alternatives in your question is completely correct.
The distribution $\Psi_{\beta}$ is as defined by equation (7) in Section 2 of your reference:
$$\Psi_{\beta}(r) = \sum_{k = -\infty}^{\infty} \frac{1}{\beta}\,\exp\left(-\pi \left(\frac{r - k}{\beta}\right)^2\right),$$

for $r \in [0, 1)$. So $\beta$ is not the standard deviation of a normal distribution (instead, the standard deviation is $\beta / \sqrt{2\pi})$. Note that $\Psi_{\beta}$ is not a normal distribution though, but a "periodization" of normal distribution.

Also keep in mind that $\bar \Psi_\beta$ is a discretized version of $\Psi_\beta$, i.e. the distribution that you get after multiplying by $p$ and rounding to the nearest integer mod $p$. This is why the introduction mentions that $\bar\Psi_{\alpha}$ has (roughly) the shape of a normal distribution with standard deviation $p \alpha$.