I've had a quick look at the paper and I can confirm that my guess in the comment above was correct.The sextics in Group I in the statement of Thm. 2.4 are those that have at most double points or triple points that do not have an infinitely near triple point. These singularities are discussed in II.8 of Barth-Peters-Van de Ven "Compact complex surfaces" (I have the old edition) where they are called simple singularities. Sometimes they are also called negligible singularities. The reason for the terminology is the following: if $C$ is a curve in a smooth surface $S$, $X\to S$ is a double cover branched on $C$ and $P\in C$ is a singular point, then $X$ has a rational double point over $P$ iff $P\in C$ is simple iff (locally near $P$) $K_Y= f^*K_X$ for the minimal resolution $f\colon Y\to X$.
In old fashioned language, these singularities "do not impose adjunction conditions" on double covers.