I'm reading Hatcher's book of Algebraic Topology where he define the notion of $\Delta$-complex. Then he puts the following diagram

And said that this shows a $\Delta$-complex structure of the torus, the real projective plane and the Klein bottle. I'm trying to figure out how this structure is. This is my interpretation.

If I have a $\Delta$-complex structure $\mathcal{A}=\{\sigma_\alpha:\Delta^{n_\alpha}\rightarrow X\}_\alpha$ on a topological space $X$ and I have a quotient space $$\pi:X\rightarrow X'=X/\sim$$ Then we have a canonical family of maps $\mathcal{A}'=\{\pi \circ\sigma_\alpha:\Delta^{n_\alpha}\rightarrow X'\}_\alpha$ and hopefully this is going to be a $\Delta$-complex structure for $X'$. So perhaps this is the $\Delta$-complex structure that Hatcher gives.

But now if we take $X$ to be a square and we give it the $\Delta$-complex structure of the figure

Then this structure is explicitly given by the maps $$\mathcal{A}=\{\sigma_{ABD},\ \sigma_{BDC}, \ \sigma_{AB}, \sigma_{BD}, \ \sigma_{DA}, \ \sigma_{CB}, \ \sigma_{DC}, \ \sigma_{A}, \ \sigma_{B}, \ \sigma_{C}, \ \sigma_{D} \}$$

And then if we take the family of maps $\mathcal{A}'$ as above, this gives a $\Delta$-complex structure in the case of the torus. But it's not a $\Delta$-complex structure in the case of $\mathbb{R}P^2$ because in this case $\pi\circ \sigma_{AD} \neq \pi \circ \sigma_{BC}$ (because if $\Delta^1=[v_0,v_1]$ we have $\bar{A}=\pi\circ \sigma_{AB}(v_0)\neq \pi \circ \sigma_{DC}(v_0) = \bar{D}$) but $\text{im}(\pi\circ \sigma_{AD})=\text{im}(\pi\circ \sigma_{DC})$ so this mess up the partition part of the definition. I think there is a similar issue with the Klein bottle.

So my questions are

How you use the diagram above to find a $\Delta$-complex structure for the real projective plane and the Klein bottle and how do you do this for other quotients of $\Delta$-complexes (such as seeing the "dunce hat" as a quotient of a triangle).