This is a very common issue in topology: The same terminology (boundary and interior) have different meaning in different branches of topology. My guess is that you already took a General Topology (Point Set Topology) class and are familiar with the notion of boundary and interior of a subset $A\subset X$ in this branch of topology.
The definition of a cell you are given is a bit sloppy. A cell is a topological space $C$ homeomorphic to the closed $n$-dimensional disk $D^n$, equipped with a certain extra structure (which has a somewhat sloppy description in the document you attached). Let us ignore the extra structure. Note that the disk $D^n$ is a subset of $R^n$. The interior and the boundary of $D^n$ are defined with respect to this embedding. In other words, the interior of $D^n$ is
$$
int(D^n)=\{x\in R^n: |x|< 1\}
$$
and the boundary of $D^n$ is
$$
\partial D^n= \{x\in R^n: |x|= 1\}= S^{n-1}.
$$
What is not immediate with this definition is that the interior and the boundary of a cell are topologically invariant in the sense that a homeomorphism $h: C\to D^n$ would send interior to interior and boundary to boundary. This is a theorem (a corollary of Brouwer's Invariance of Domain theorem). Once you have this invariance, you can talk about the boundary and interior of a cell $C$, which is regarded as an abstract topological space, not embedded anywhere.
Now, a (nondegenerate) line segment $pq$ in $R^n$, $n>1$, is still a 1-cell simply because it is homeomorphic to $D^1=[-1,1]$. Of course, the interior of $pq$ in the sense as above is not the same as the interior of $pq$ as a subset of $R^n$ (the latter is empty).
If you are bothered by this "double speak", you can refer for yourself to the "interior" defined above as "cell-interior" and the boundary as the "cell-boundary". At the same time, you can call the interior of a subset $A\subset X$ in the sense of general topology as "topological interior".
Hope it helps.
Hatcher describes how CW-complexes are constructed:
Start with a set $X^0$ having the discrete topology.
Construct inductively skeleta $X^n$ by attaching $n$-cells to $X^{n-1}$. Here $X^{n-1}$ already has a topology and $X^n$ is defined as a suitable quotient space of $X^{n-1}\bigsqcup_\alpha \mathbb{D}_\alpha^n$.
This construction produces a sequence of spaces $X^0, X^1, X^2,\ldots$ such that $X^{n-1}$ is a subspace of $X^n$.
If this process stops at some finite $N$, then we have a topology on $X = X^N$. In that case trivially $A \subset X$ is open iff $A \cap X^n$ is open in $X^n$ for all $n$. Just note that $X^n = X$ for $n \ge N$.
If the attaching process continues ad infinitum, then we have a topology on each skeleton $X^n$, and these toplogies are compatible in the sense that $X^{n-1}$ is a subspace of $X^n$ for each $n$. Howewer, we do not have a topology on $X = \bigcup_{n=0}^\infty X^n$. That is why we give the space $X$ the weak topology defined as in (3). This definition is relevant only in case that the attaching process continues ad infinitum.
Edited on request:
Suppose we have an ascending sequence of topological spaces $X^0, X^1, X^2,\ldots$ (ascending means that for all $n > 0$ we have $X^{n-1} \subset X^n$ and that the topology on $X^{n-1}$ is the subspace topology inherited from $X^n$). Then let $X = \bigcup_{n=0}^\infty X^n$. Which topology do we introduce on $X$ in order that all subspaces $X^n$ receive their original topology? In general there are many ways to do that, but the standard approach is to define $A \subset X$ open in $X$ iff $A \cap X^n$ is open in $X^n$ for all $n$. This is the final topology with respect to the system $\{X^n\}$. In the context of CW-complexes it is also denoted as the weak topology. This has historical reasons. See Confusion about topology on CW complex: weak or final?
Let us prove that if $X$ is endowed with the final topology, then all subspaces $X^n \subset X$ have their original toplogy:
Let $U \subset X^n$ be open in the subspace toplogy. Then $U = A \cap X^n$ with some open $A \subset X$. But by definition of the final topology $A \cap X^n$ is open in $X^n$ with its original toplogy.
Let $U = U_n \subset X^n$ be open in the original topology. Using the fact that each $X^{k-1}$ is a subspace of $X^k$, we can recursively construct open $U_k \subset X^k$, $k \ge n$, such that $U_{k+1} \cap X^k = U_k$. Moreover, for $k < n$ define $U_k = U_n \cap X^k$ which is open in $X^k$. By construction we have $U_k \cap X^m = U_k$ if $k \le m$ and $U_k \cap X^m = U_m$ if $k > m$. Then $A = \bigcup_{k=0}^\infty U_k$ is open in $X$: In fact, $A \cap X_m = (\bigcup_{k=0}^\infty U_k) \cap X_m = \bigcup_{k=0}^\infty U_k \cap X_m = \bigcup_{k=0}^m U_k \cup \bigcup_{k=m+1}^\infty U_m = U_m$ which is open in $X^m$. This also shows $A \cap X^n = U_n = U$, i.e $U$ is open in the subspace topology.
If the sequence $X^0, X^1, X^2,\ldots$ stabilizes, i.e. if we have $X = X^N$ for some $N$ (which is the same as $X^n = X^N$ for $n \ge N$), then there is no need to introduce a new topology on $X$ since $X^N$ already has one. The above proof shows nevertheless that $A \subset X$ is open iff $A \cap X^n$ is open in $X^n$ for all $n$. However, in that case it is trivial: If $A \cap X^n$ is open in $X^n$ for all, then $A = A \cap X = A \cap X^N$ is open in $X = X^N$. Conversely, if $A \subset X^N$ is open, then $A \cap X^n$ is open in $X^n$ for all $n \le N$ because $X^n$ is a subspace of $X^N$ and open in $X^n$ for all $n > N$ because $X^n = X^N$ and $A \cap X^n = A$.
Best Answer
Hatcher defines in Standard Notations on p.xii
The phrase "open $n$-cell" is synonymous with "$n$-cell", but typically avoided as it can be misleading (an open $n$-cell in a topological space need not be an open subset thereof). The one instance of this phrase in Hatcher's book is presumably just a slip of the tongue. The one appearance in Husemoller's book is in the introduction, where he does not define the terminology, but redirects the reader to J.H.C. Whitehead's paper Combinatorial Homotopy. I. There, at the start of Section $4$, we find
This informs us that "open cell" and "cell" are the same concept and the adjective "open" is considered negligible.
The term "closed cell" is ambiguous. It does not seem to appear in Husemoller's book. In Hatcher's book, there is one instance on p.535. He calls a CW-complex regular if the characteristic maps can be chosen to be embeddings. In this case, the closures of the $n$-cells of this CW-complex are homeomorphic to the (closed) disk $D^n$ and he calls them "closed cells". However, in other contexts, one might want to refer to the closure of any cell in any CW-complex (which is not always homeomorphic to a closed disk) as a "closed cell" (this is done e.g. in Bredon's Topology and Geometry).