For any crystal, the First Brillouin Zone is found using the Wigner-Seitz construction for the reciprocal lattice. The high-symmetry points are labeled by certain letters mainly as a convention--like you said Gamma for (0,0,0) etc.
The important thing to realize as far as the group theory, is that the group of the wavevector at the Gamma point has the full point group symmetry of the real space lattice. However, certain high symmetry wavevectors, labelled by the different Greek letters, are subgroups of this group. That is, only a certain number of the symmetry operations of the point group at the Gamma point (rotations, mirrors, etc.) will leave the new high symmetry point invariant. Thus the benefit of symmetry is that you only have to consider an even smaller region of the BZ to get all of the reciprocal space information about the crystal. This is called the Irreducible Brillouin Zone, and paths along the high symmetry points of the IBZ are used as the x-axis in band structure diagrams.
Use this website to explore different Brillouin zones: http://www.cryst.ehu.es/
This paper gives a thorough treatment of many Brillouin zones: Setyawan, Wahyu, and Stefano Curtarolo. "High-throughput electronic band structure calculations: Challenges and tools." Computational Materials Science 49.2 (2010): 299-312.
The following Book is a great resource which has tables of the high symmetry points in the FBZ and their point groups:
Dresselhaus, Mildred S., Gene Dresselhaus, and Ado Jorio. Group theory: application to the physics of condensed matter. Springer Science & Business Media, 2007.
The first BZ should contain one lattice point, the second BZ two, and the third BZ three, etc. Figure 1 is correct because it is exactly the case. The third BZ in Figure 2 contains 4 lattice points, so it is not correct.
Best Answer
I think there is no general answer to that. Just try and include all important symmetry directions you can think of.
For example, in the case of the Brillouin zone shown in your left picture, how about the path K'-K-Gamma-K'-M-K-Gamma-M? (I know, the path Gamma-K now is included twice, but I couldn't think of any other possibility.)