I'm very new to algebraic topology and am trying to wrap my head around CW complexes. My main sources are Allen Hatcher's Algebraic Topology and these online notes by Soren Hansen. I outline my current understanding below and would be very grateful if you could take a look and point out any mistakes or details that I've skirted over.
For each $n\in\mathbb{Z}_{\geq0}$, we denote the closed unit ball in $\mathbb{R}^n$ (w.r.t. the Euclidean norm) by $D^n$. First of all, it is important to note that since $\text{Int}(D^n)$ is homeomorphic to whole of $\mathbb{R}^n$, we have that $\text{Int}(D^m)$ and $\text{Int}(D^n)$ are homeomorphic $\iff$ $\mathbb{R}^m$ and $\mathbb{R}^n$ are homeomorphic $\iff$ $m=n$.
We define a cell to be a topological space $X$ that is homeomorphic to $\text{Int}(D^n)$ for some (necessarily unique by the previous paragraph) $n\in\mathbb{Z}_{\geq0}$, which we call its dimension. We refer to a cell of dimension $n$ as an $n$-cell.
A cell decomposition of a topological space $X$ is a partition of $X$ into a family of cells $\mathcal{E}\equiv\{e_\alpha\}_{\alpha\in I}$. (By a 'partition', I mean that the cells $e_{\alpha}$ are non-empty, mutually disjoint and cover the whole of $X$.) The $n$-skeleton of $X$ is the union of all cells of dimension no greater than $n$, which we denote by $X^n$.
We now define a CW complex to be a Hausdorff space $X$ with a cell decomposition $\mathcal{E}\equiv\{e_\alpha\}_{\alpha\in I}$ satisfying
(i) For each $n$-cell $e\in\mathcal{E}$, there exists a continuous map $\Phi_e:D^n\to X$ that maps $\text{Int}(D^n)$ homeomorphically onto $e$ and the boundary $\partial D^n\equiv S^{n-1}$ (not necessarily homeomorphically) into the $(n-1)$-skeleton $X^{n-1}$.
(ii) The closure $\overline{e}$ of each cell $e\in\mathcal{E}$ intersects only finitely many other cells in $\mathcal{E}$.
(iii) A subset $V$ of $X$ is closed if and only if $V\cap\overline{e}$ is closed for each cell $e\in\mathcal{E}$.
My main concern here is that there are many different definitions that I have seen which do not look alike, and it feels like I'm missing something. Is there a more intuitive way of thinking about CW complexes? Thanks for your help. :)