I'll address all questions individually, but I'll make no attempt to infer a general question.
We use the definite article in front of a noun when we believe 'the reader' knows exactly what we are referring to
- My first question, am I right in thinking that this is a general statement?
That describes the definite article, definitely. Generality is not the correct aspect, because as you noted, there are different ways to express it. Ironically in that sentence, the reader does not exactly know that which is referred to, otherwise the explanation would not have had been necessary.
- what if we use 'readers' in place of 'the reader'?
That would be correct in some sense, but the noun would not be definite. The difference is the same as between the or a reader. In "a reader", the singular is determined, but the referent is explicitly not defined. One could ask for which reader was meant. Likewise, in "readers" the plural is determined, but in "the readers" the referent group is definite. Talking about all the readers assumes that the distinction is definite. The indefinite reference is less specific, it implies a specific distinction within the group of the readers. It's rather philosophical, if not mathematical.
Both versions are used. "the" seems formal, to me.
- I'm asking this question because if this is a general statement then if I use 'a reader' or 'readers' in place of 'the reader' then it should all mean the same, i.e., all of them should be expressing 'any reader' just like the examples above (1 to 4). Right?
No, that is not entirely correct. 2 and 3 talk about one out of many, as if the sentence wasn't true for all the other members of the group. Concerning plural vs singular, notice e.g. how I tried to use singular in the previous sentence, "the sentence/group", to achieve a general statement, because your sentence 2 is obviously not true for the groups of the other sentences. This seems to be the only viable alternative of those four (sentences, a sentence, the sentence and the sentences).
(1) Banks are financial institutions where 'a lender' meets 'a borrower'.
(2) Banks are financial institutions where 'lenders' meet 'borrowers'.
- I guess this (1) is also a general statement about banks. So can I phrase this sentence like this (2) without any change in meaning?
Yes, you can. Actually you improved it, because a bank with a single lender and borrower would be exceptional. Although, in the example (1), the indefinite singular obviously implies a plurality, simply from context. Whereas, the other way around, a plural form meaning at least one is not wrong either, to my mind, but this understanding might lead to confusion because not everyone agrees with this interpretation.
- In sentence (1), 'a lender' expresses just one lender or any lender? In sentence (2), 'lenders' expresses more than one lender or any lender in general?
(1) expresses a one-to-one relationship, which can be extended to mean any of one-to-many relationships ...
Edit: The article "a" can mean e.g. "one" or "any", "anyone" and much more, depending on context. It is unspecific by design, hence indefinite.
Context matters, ie. the expectation of the writers and the readers. It is unspecific on purpose and could mean anything. With laws, e.g. the expectation is that they are general. With specific questions, the expectation is that the answer would be as precise as possible.
"The dog is an animal" is the correct way to frame a definition (hence definite article), but the indefinite article is commonly accepted, too. With loyal dogs the picture is different, because the saying all dogs were loyal would be a gross over-generalization. The commonly accepted answer says "a ... is ..." is a generalization.
See also here and here and here and here and here and here.