Topologically, we can turn a rectangle into a Möbius strip by gluing the edges so that the directions of the arrows match. I list all the possiable edge side-identifications for a rectangle. What I mean edge side-identifications is how one can glue the edges geometrically. Therefore, there are 9 different side-identifications as described in the following figure.
We can easily see that:
(1): the original ratangle without gluing any edges;
(2): a cylinder by gluing the same colored edge (here is red) with matched direction;
(3): a Möbius strip by gluing the red colored edge with matched direction;
(4): a cylinder by gluing the same colored edge (here is blue) with matched direction;
..., etc.
When the retangle becomes a square, then we will have additional edge side-identifications.
Question: Is there any terminology or mathematical name for the above listed classification of side-identifications? Are there literature references?
I very much hope for some hints or further literature references on these questions. Thank you in advance!
While there is no standard terminology in this specific situation, nor can I point you to a specific reference, there is a well known descriptive classification using topological terminology. Namely, one may reduce this problem to classification of graphs embedded in surfaces. Here's how that reduction is carried out.
The first point to make, though, is that it is natural to consider side-identifications under an equivalence relation: given side-identifications defined on two quadrilaterals $Q_1$, $Q_2$, they are equivalent to each other if there exists a homeomorphism $h : Q_1 \mapsto Q_2$ taking vertices to vertices and sides to sides, such that for any two points $x,y \in Q_1$, $x$ and $y$ are identified if and only if $h(x)$ and $h(y)$ are identified. If you use this equivalence relation you will see that the distinction between rectangles and squares is irrelevant, and furthermore the distinction between two side-identification patters of a square that differ by a symmetry of that square are irrelevant. Also "colors" such as blue and red are irrelevant, all that matters is which side-pairs are identified and which directions on each side-pair are identified.
With this in mind, you should then be able to see that up to equivalence there are somewhat fewer side-identification patterns of quadrilaterals than what you counted (I get a total of 9 equivalence classes, although I have not double checked that). For example, your patterns (2) and (4) are equivalent, also (3) and (5) are equivalent, also (7) and (8) are equivalent.
As for further topological terminology to describe this equivalence relation, consider two polygons $P_1,P_2$ with side-identifications specified, and consider the quotient surfaces $S_1,S_2$. Consider also the image under the quotient map $P_i \mapsto S_i$ of the boundary of $P_i$, which is a graph in $S_i$ denoted $G_i$. One can prove that $P_1$ is equivalent to $P_2$ if and only if there is a homeomorphism $S_1 \mapsto S_2$ which restricts to a graph isomorphism $G_1 \mapsto G_2$. For example, regarding equivalence of (2) and (4), there is a homeomorphism from the quotient annulus of (2) to the quotient annulus of (4) taking the red colored edge to the blue colored edge.
So the question of classifying side-identifications up to equivalence can then be converted into a more topological classification problem, namely the classification of pairs of the form $(S,G)$ such that $S$ is a compact surface and $G$ is an embedded graph (with some further restrictions on the pair, in order to restrict attention to pairs that come from polygon side-identifications).