I didn't notice this before, but the link provided in the question can already be used to visualize any 2D Brillouin zone. Since you can define the ratio of b/a, and the angle between b and a, then you can define any 2D Bravais lattice. There are only 5 possibilities. The "hexagonal" and "square" options are just there to conveniently show two of them.
The Brillouin zone only depends on the Bravais lattice vectors. Any additional complexity in the basis of this lattice (i.e. the specific arrangement of tiles) is irrelevant for the shape and size of the Brillouin zone. However, in real diffraction pattern measurements, the basis determines the intensity of each point in the reciprocal lattice by the corresponding structure factor.
I drew some of the possible Bravais lattice unit cells for the tile patterns included in the question: