Ok, so based on the discussion in the comments, maybe I should put this into an answer.  I think the confusion comes from the phrase

*every codimension 1 point of $\text{Sing }X$.*

What the authors Kollár and Kovács mean here is to consider 

*every point of $\text{Sing }X$ that is also a codimension $1$ point of $X$.*

I agree that this could be interpreted in other ways, but this is what the authors mean (I'm sure Sándor will agree if he sees this).

Given this, and that $X$ is normal over $\mathbb{C}$ (or any field of characteristic zero), it is easy to see that every cyclic cover is a *ramified cover* in the sense of Kollár-Kovács (as I think you already see).  As pointed out in the comments, if $X$ is non-normal, or if we are working in characteristic $p > 0$, life becomes more complicated.

<h3>Your particular situation</h3>
You had $X$ with canonical singularities and $\widetilde{X}$ a ramified cyclic cover.  Then it is easy to see that $(X, -\text{(Ramification Divisor)})$ also has canonical singularities (notice we have a non-effective divisor here).  For a proof simply see Kollár-Mori 5.20(3).