Many thanks for your nice answer and a reference. In the cases $D = 4$, $(2,2)$ and $D = 6$, $(3,3)$ there are also examples of flat spin manifolds $M^2$ and $M^3$, respectively, where $M = \mathbb R^{(1,1)}_{*}/Z_2 $. The orginal problem for me was to find examples of: i) 5-dimensional Riemannian Ricci-flat spin manifold with only one parallel spinor and ii) 6-dimensional pseudo-Riemannian Ricci-flat spin manifold with either only one parallel spinor or two parallel spinors of opposite chiralities. So the chance is small, but it will be analysed using your classification.

Your both examples are correct.