In Uniqueness, Familiarity, and the Definite Article in English (pdf link) by Betty Birner, University of Pennsylvania, and Gregory Ward, Northwestern University, the authors talk about the use of the definite article with mass and plural nouns. They mention that in both cases, the noun phrase does not have to refer to all of the things mentioned in the definite noun phrase.
For example, see sentence 12:
When I was traveling through Switzerland last year I took a beautiful photograph of the mountains.
The mountains does not refer to (a) all the mountains in Switzerland, or even (b) "every mountain in Switzerland seen by the speaker". It refers to an "undifferentiated subset" of the mountains. This is the same usage as the posts there and the books in the NYPL.
The use of the in a NP with plural count noun (whether mountains, posts, books) does not have to, and sometimes absolutely cannot, refer to all of the things.
The same is true of the bare plural versions if the same noun phrases. Here, the reference is indefinite, and that includes being indefinite as to number. See also the ELL question Necessity of a definite article before the antecedent of relative pronoun for another natural example of the bare plural noun in the sentence
Johnson was helping her country win the space race, calculating trajectories that got spacecraft in and out of space.
The noun phrase trajectories that got spacecraft in and out of space is indefinite as to number. The use of posts there shows the same indefiniteness regarding number. It could refer to all posts but that is not the intent. The intent is to be indefinite.
Using the in the posts there makes the noun phrase definite, but here, as per Birner & Ward, the definite noun phrase does not have to refer to all the posts. What is important, per Birner & Ward is context and inferring the speaker's intent.
Thus, the does not have to refer to all the posts or all the books. The job of the is to make a definite reference instead of an indefinite reference. For all we know, posts could refer to all the posts, but since you make an indefinite reference we can't be sure about the number of posts you mean.
The posts by itself does not refer to all of them. Consider a lady who just sat down with a magazine. Her husband asks her what she is doing. She replies that
she is looking at the ads in the magazine.
She is just making a definite reference; it does not correspond to all the ads, which is what she might say if she were determined or intended to look at them all. Or a boy at high school. He has just started dating. Who does he date?
He dates the girls at his school.
Does this mean he will date all the girls. No.
As a native speaker, the use of the posts in your sentence is what I would use. And I and your Lang-8 proofreader instinctively felt that this was more idiomatic in this context. In one sense that's the entire answer. John Lawler has said in a comment to an ELU post and I paraphrase: English articles are idiomatic and one needs to (almost?) take them on a case to case basis.
On the other hand, you could say posts, but without the definite article it sounds vague; it does this because it is not a definite reference. It is not a question of either/or, but which kind of reference do you wish to make. To the native speaker who corrected you and to me, a definite reference sounds more idiomatic in this context, but it does not rule out saying just posts.)
Here's an example that is similar to yours:
The prices were high at Nieman Marcus. Well except for the fur coats, which were on sale for 50% off.
I just used the fur coats. This is a definite reference. And I can use the because I think you can identify which ones I am talking about. Namely, the ones at Nieman Marcus. But I don't have to use the.
I could have just said Well, except for fur coats, which were on sale for 50% off. This is an indefinite reference (with the zero article). It still means the fur costs at Nieman Marcus: this is clear from context. But otherwise it is not marked as definite.
This is comparable to your OQ. I can make a definite reference if I want. Or I can make an indefinite reference if I want. In this example, making a definite reference seems to be more natural and idiomatic here, and in your context about the posts.)
Note also you use the progressive past tense (was reading), which describes an action in progress. You were reading the posts there when one of them caught your attention, or you were interrupted.
I was tasting the wines at the the local vineyard when my wife called me on my cell phone.
I was tasting the wines at the local winery when I found one I decided to buy. Notice I don't intend to taste all of them, but when I find something I like, I will buy it.
(I would avoid the indefinite mass noun wines here because, like posts it just sounds too indefinite. Wishy washy.)
As for the NYPL, a native listener would assume you mean the books carried by the library. And unless you say otherwise (my books) this is a natural assumption. Just like I am going to the office. Unless I stipulate otherwise, you can assume I mean my office or whichever one I usually go to. The key is that I expect you to be able to identify which office I am talking about.
Again, the books does not mean all the books.
On using the definite article or not, see my answer on ELU to Are there any simple rules for choosing the definite vs. indefinite (vs. none) article?