Consider the vertex configuration $(3.4)^3$ which yields the alternated octagonal tiling
and can be found also here:
I have no idea how the fundamental domain, resp. primitive cell of this tiling looks like, how to construct, and how to depict it (from which the tiling can be created by some translations in the hyperbolic plane). Can someone please be of help?
The wikipedia table has a column "Fundamental triangles". Those little triangles are the fundamental domains (basic cells) used to construct the tiling by reflections across the fundamental triangle's edges. Compositions of reflections across edges then give translations and rotations.
Note that even though the triangles look differently due to the hyperbolic metric, they are actually all congruent copies.
If you are interested in the underlying symmetry group of those tilings, check for the term reflection group or Coxeter group. The parameters (4.3.3), (4.4.3) etc. specify the relations of the reflection group generated by the three reflections across the edges of the frundamental triangle.