Gluing hexagons to get a locally CAT(0) space
The third example is a spine of the Gieseking manifold. The existence of a spine of this sort follows from a result of Iain Aitchison and the fact that the Gieseking is make of a single regular ideal tetrahedron. I think one could get this directly from the combinatorial description, but I also checked that it's a 1-relator group that fibers with fiber with the same monodromy as the Gieseking manifold.
The group has a presentation $\langle a, b | a^2b^2a^{-1}b^{-1} \rangle$. Letting $u=ab$, we can change to a presentation $ \langle a , u | a^2ua^{-1}ua^{-1} u^{-1} \rangle$. This has abelianization $\mathbb{Z}$ with $a\to 1, u\to 0$. Hence the kernel is generated by $u_i=a^i u a^{-i}$, and the relator shows that $u_0=a(u_1 u_0)a^{-1}, u_1=au_0a^{-1}$, hence this is a free-by-cyclic group with the same monodromy as the Gieseking.
The fourth complex has fundamental group a 1-relator group $\langle a, b| a^2b^2a^{-1}b\rangle$. One may see also that this is a rank-3-free-by-cyclic group. It also has an automorphism given by $(a,b) \mapsto (a, a^{-1}ba )$. Making the substitution $u=b^3a$, one gets a presentation $\langle b,u| bub^{-3}ub^2u^{-1}\rangle$. If $u_i=b^iub^{-i}$, then one has the relations $u_0=b^{-1}(u_0 u_{-2}^{-1})b, u_{-1}=b^{-1}u_0b, u_{-2}=b^{-1}u_{-1}b$, and hence this is the mapping torus of the free group automorphism $(u_0,u_{-1},u_{-2}) \mapsto (u_0u_{-2}^{-1},u_0,u_{-1})$.
Since the ranks of the kernels of the maps to $\mathbb{Z}$ are different, the groups are not isomorphic.
I'm guessing that you can find these examples in the literature on 1-relator groups, but I'm not sure if they have names in that context.
The fourth example was studied by Brady and Crisp in their CMH paper CAT(0) and CAT(-1) dimensions of torsion-free hyperbolic groups, so it would be reasonable to call its fundamental group the "Brady--Crisp group". (Brady and Crisp also note that it belongs to a family studied by Haglund and Ballmann--Brin.)
They study a one-parameter family of CAT(0) metrics on the complex, and prove the very nice fact that any CAT(-1) model for this group has to have dimension at least 3. (And they exhibit a 3-dimensional CAT(-1) model.)