I think the biggest difference between Rips complex and Cech complex is that Rips complex includes the "triangle" when it has all the "edges". And the reason I thought why Rips complex automatically includes the triangle is as below.
When all the edges are included, then we can draw lines between any arbitrary points $p, q$ on those edges, because the distance between any points of those edges will be less than t as well. And these lines between arbitrary $p$ and $q$ will fill the inside of the triangle.
But because Rips complex is an abstract simplicial complex, I cannot draw lines without geometric realization. Then I don't know how I can explain the reason Rips complex includes the triangles. Is there anyone who can explain the reason?
(+ This made me look into the concept of abstract simplicial complex again. And I don't think I get the concept clearly. For example, if I make an abstract complex $$A = \{a, b, c, \{a, b\}, \{b, c\}, \{c, a\}, \{a, b, c\} \}$$ how should I understand $\{a, b\}$? It's not an edge obviously. But then, is there no connection between vertices $a$ and $b$?)
It is known that for a fixed $\delta>0$ the Čech complex is a subcomplex of Rips complex. And indeed, the main difference is that the Čech complex does not have to contain $n$-dimensional triangle ($n\geq 2$) given that it contains all of that triangle faces. The Rips complex indeed has this property.
That's incorrect. Abstract simplicial complexes do not have "arbitrary points on edges", they do not have notion of "distance", they do not have "lines" and you don't "fill" anything in them. They also don't have "inside". All of those terms come from geometric/metrical background, and they do apply to geometric simplicial complexes (to some extent), but not to abstract.
Of course there is a nice way of constructing a geometric simplicial complex from an abstract one, known as geometric realization, but this construction won't help you much with the problem you are dealing with. That's because the Čech/Rips complex arises from some metric space, and the geometric realization is very loosely related to that metric space.
I've explained this for you here: Can there be arbitrary vertices on the edge of an abstract simplicial complex?
So first of all, lets fix your notation. The $A$ you wrote above requires some intermediate brackets and missing vertices (you don't distinguish between points and abstract simplexes of dimension $0$ which is formally incorrect, and a bit confusing IMO). The abstract simplicial complex $(X,S)$ is a pair, where $X$ is a nonempty set and $S$ is a collection of nonempty subsets of $X$ such that if $D\in S$ and $D'\subseteq D$ is nonempty then $D'\in S$. So in your case it should be written as
$$X=\{a,b,c\}$$ $$S=\big\{\{a\},\{b\},\{c\},\{a,b\},\{b,c\},\{c,a\},\{a,b,c\}\big\}$$
And in that situation $(X,S)$ is indeed an abstract simplicial complex. Elements of $X$ we call vertices, while elements of $S$ we can call abstract simplexes (or $n$-dimensional triangles).
Now, how should you understand $\{a,b\}\in S$? You say its not an edge. Well, this is just a matter of naming. I can call it edge if I want, and in fact when we pass to the geometric realization it does correspond to an edge. And so it is not a bad naming.
Generally $\{a,b\}$ is just a set of two elements. It does not have intermediate points, it does not have topology or metric or geometry. It captures a very basic property of an edge: that two ends of an edge fully describe it. Analogously $\{a,b,c\}$ capture a basic idea that every triangle can be fully described by its three vertices. And so in the "abstract" setup, we don't deal with those edges, triangles and other geometric stuff, we simply say "hey, from now on edge will mean a set of two elements $\{a,b\}$". This looks like a basic, even primitive idea. But it turns out that many properties of complexes can be derived simply by looking at those abstract representations. Plus it is easier for a computer to deal with finite sets instead of infinite.
Sometimes it is useful to think about abstract simplicial complexes in terms of their geometric realization. You may build some intuition in that way. But ultimately, when it boils down to proofs, abstract simplicial complexes are nothing else than what the definition says. There is no metric, no geometry, no distance, no intermediate points. Just points and their combinatorical relationship.