One way to classify the ways that a finite group can act on the regular hexagonal torus $H$ is to first lift the action to the plane, then name the so-called wallpaper group $W$, and then determine whether (and in what ways) the fundamental group of $H$ can arise as a subgroup of $W$. Using Dror Bar-Natan's illustrated list of the 17 wallpaper groups, the 3 that possess a 90 degree rotation are impossible in this context. I think that the others are all possible, but there are subtle difference in group structure because of the distinction between a reflection and a glide reflection. I.e., $W$ always has a translational subgroup $T$, which is a normal subgroup, and a rotational quotient group $R$, but it can be a non-split extension of $R$ by $T$, not just a semi-direct product.
Your list of candidates is not quite complete. In the case that the orbifold quotient (of either the torus $T$ or the whole plane) is a Klein bottle, it means that you have glide reflections but no reflections. In this case the finite group acting on $H$ can be a non-split central extension of a dihedral group by a cyclic group. Besides, even after listing the finite groups that can act on $H$, a particular group in the list can act in more than one way, with subtle differences. For instance I think that there are 5 ways that the group with 2 elements acts on $H$, because the quotient can be a Klein bottle (in two ways), an annulus (again in two ways), or a sphere with four cone points of order two (in one way).