Let $(X,\Delta)$ be a log pair (we assume the coefficients of $\Delta\leq 1$, but could be negative rationals), and I use the definitions in the book of "Birational Geometry of Algbebraic varieties" (Page 56, Def. 2.34; Page 58, Def. 2.37) for "terminal, canonical, klt, plt, lc, dlt" singularities.

My first question is: which of these six singularities do not depend on the resolution of singularities?

To be precise, this means: if one happen to have a log resolution: $Y \to X$, and the discrepancy for this resolution satisfy the requirement for the corresponding singularities, then for all the log resolutions, the discrepancy satisfy the requirement.

I try to use the standard argument that one dominant two resolutions by the third one. However, I always feel I might miss something (because the argument goes through when the coefficients of $\Delta > 1$, in which case, all the statement are false), below is what I did:

Suppose we try to "show" lc doesn't depend on the resolution. Let $\pi: Y_1 \to X$ be the log resolution, and the coefficients of the exceptional divisors $E_1$ are all $\geq -1$ in the log reamification formula. Let $\theta: Y_2 \to X$ be another log resolution. Finally, let $p:W \to Y_2, q: W \to Y_1 $ be a common resolution where the exceptional divisors and the strict transform of $\Delta$ (which I denote by $\Delta_W$) have snc. Then, we have $$K_W + \Delta_W \sim q^*\pi^*(K_X+\Delta) + q^* E_1 + F_1,$$ where $E_1, F_1$ are some exceptional divisors of $\pi, q$ respectively. We can write $$q^*E_1 = q_*^{-1}E_1 + Exc_q(q^*E_1)$$ where $ q_*^{-1}E_1$ is the strict transform of $E_1$, and $Exc_q(q^*E_1)$ belongs to the exceptional divisor of $q$. One can do the same thing for $W \to Y_2 \to X$, and putting them together we have $$q_*^{-1}E_1 + Exc_q(q^*E_1) + F_1 \sim \theta_*^{-1}E_2 + Exc_p(p^*E_1) + F_2.$$ Because every divisor involved in the above expression are exceptional divisors of $p \circ \theta = q \circ\pi$, the numerically equivalence implies equality of divisors. Moreover, because the coefficients on the left are all $\geq -1$ (I am a little worry about $Exc_q(q^*E_1)$, but I think the assumption on the snc of divisors would imply its coefficients $\geq -1$ ), this implies the same thing on the right, and particularly for $E_2$.

My second question is:

Assume the coefficients of $\Delta$ is in $[0,1]$, then what are the differences between "plt" and "dlt"? I didn't realize the subtlety of $dlt$ compared to other type of singularities...