I'd like to ask for help in understanding the definition of the $\Delta$-complex structure in Hatcher's book. There is the following definition:
A $\Delta$-complex structure on a space $X$ is a collection of maps $\sigma_\alpha : \Delta ^n \rightarrow X$, with $n$ depending on the index $\alpha$, such that:
(i) The restriction $\sigma_\alpha | \mathring {\Delta} ^n$ is injective, and each point of $X$ is in the image of exactly one such restriction $\sigma_\alpha | \mathring{\Delta}^n$.
(ii) Each restriction of $\sigma_\alpha$ to a face of $\Delta^n$ is one of the maps $\sigma_\beta : \Delta^{n-1} \rightarrow X$. Here we are identifying the face of $\Delta^n$ with $\Delta^{n-1}$ by the canonical linear homeomorphism between them that preserves the ordering of the vertices.
(iii) A set $A \subset X$ is open iff $\sigma_\alpha^{-1}(A)$ is open in $\Delta^n$ for each $\sigma_\alpha$
So, what confuses me: suppose we're given two maps $\sigma_1 : \Delta^2 \rightarrow X$ and $\sigma_2 : \Delta^2 \rightarrow X$, and the images intersect each other in a face $\Delta^1$. Now I can restrict $\sigma_1$ and $\sigma_2$ to the face $\Delta^1$ (by definition, I also take a composition with canonical linear homeo). However! The obtained restrictions can be different, because $\sigma_1 | \Delta^1$ and $\sigma_2 | \Delta^1$ can behave differently on the same face (like a curve might have different parametrizations). This means that (i) fails: we might have a point $x \in \Delta^1$ which is in the image of two different $\sigma_i | \Delta^1$.
I suppose that there should be some kind of equivalence relation, and the images of simplices can be intersected only by faces preserving orientations.
You are right that axiom (i) rules out the situation you describe. I don't know why you think that's a problem, though. This is a feature, not a bug: it guarantees that if two simplices of our complex share a face, then that face is parametrized the same way in both of them.
In other words, a $\Delta$-complex isn't just a union of simplices which overlap on their faces. These overlaps are required to be consistent with the parametrizations of the faces in a strong way.
It may be instructive to consider how a classical finite simplicial complex is always a $\Delta$-complex by this definition. Suppose you have a finite simplicial complex: that is, some collection of faces of a simplex $\Delta^N$, where if a simplex is in your collection then all faces of it are also in your collection. Such a space can be made a $\Delta$-complex by fixing an ordering of the $N+1$ vertices of $\Delta^N$ and using this ordering to parametrize every simplex in your collection. The fact that we're using the same ordering of the vertices to parametrize every simplex means that when two simplices share a face, they parametrize it in the exact same way, as axiom (i) requires.