Question about Set Theory notation (from preliminaries of Hubbard & Hubbard, Vector Calculus)

Question

In section 0.3 of the preliminaries for Hubbard & Hubbard's Vector Calculus text, there is an example describing the concept of indexed set notation which I'm currently having trouble interpreting. The example states that: for the subset $ \ell_{n} \subset \mathbb{R}^{2} $, where $ \ell_{n} $ is the line of the equation $ y = n $, we have $ \bigcup_{n \in \mathbb{Z}} \ell_{n} $ as the set of points in the plane where the ( y )-coordinate is an integer. In the margin, this is compared to the set {$ \ell_{n} \mid n \in \mathbb{Z} $} which is described as a set of lines rather than a set of points, highlighting that the two sets are different because of this. I'm confused as to why they are interpreted differently when both examples are sets of $ \ell_{n} $, which is defined to be the line $ y = n $ in the beginning. Why would this defined to be a set of points in the first example and a set of lines in the second example? My guess would be that it would have to do with a different number of layers of embedded sets, but I'm still confused by the interpretation of points versus lines in the plane. Thanks in advance for the input.