The following book was (and still is) a valuable resource together with Munkres Topology:
Aspects of Topology by C.O. Christenson and W.L. Voxman is an easy to read, instructive text about general topology containing a lot of nice graphics.
This AMS review might be useful.
In addition and independent to text books as above I would like to put the focus on:
Counterexamples in Topology by L.A. Steen and L.A. Seebach. This is a great resource to look for topological spaces having specific properties and to look for topological properties and their relationship. It contains extensive charts of terms like compactness and the relationship of their different flavors. See also this Wikipage.
In general, definitions containing "$\dots$" means that everybody can intuitively understand them, but they don't provide details for a formal and rigorous definition of the object.
What is the problem? If you want to prove some properties about an object defined by means of "$\dots$", since you don't have a rigorous definition, you don't know exactly how to formally prove it, even though you intuitively understand what you have to prove: your proof will be inevitably hand-waved, and this could be a way to overlook some important and unexpected details (for instance, the fact that you need some further hypothesis to prove the desired property).
So, your definition of the set of positive integers as $\mathbb{Z}_+= \{1, 1+ 1, 1+ 1 + 1, \dots\}$ is perfectly understandable but then if you want to prove something about $\mathbb{Z}_+$, what do you do? Thanks to the rigorous and formal definition of $\mathbb{Z}_+$ as the smallest inductive subset of $\mathbb{R}$, it is clear which are the elements of $\mathbb{Z}_+$ and how you can use and refer to them.
By the way, there are many different but equivalent ways to define $\mathbb{Z}_+$ formally and rigorously. As @Théophile pointed out in his witty comment,
topologists love unions and intersections
so this is a possible reason why, in his topology handbook, Munkres defined $\mathbb{Z}_+$ as the intersection of some subsets of $\mathbb{R}$. This is not only a joke, but also due to the fact that this definition (among all the possible ones) is maybe the handiest one to deal with positive integers in a topological context, where you usually cope with intersections and unions.
Best Answer
Only Munkres can answer your question "why"? Anyway, his definition is correct and your definition is correct, so it is a matter of taste which you prefer.
But do you really believe that your definition is more transparent? You first define the set $S_n$ and then observe that it is empty if for $n = 1$. This step is nothing else than thinking about the two cases $n >1$ and $n = 1$. And, by the way, it would be easier to define $S_n = \{ x \in \mathbb Z \mid 1 \le x \le n\}$. Then $S_0 = \emptyset$ and $S_n = \{1,\ldots,n \}$ for $n \ge 1$. Using $S_n$ is close to what Munkres does: You have the empty set and the nonempty sets $\{1,\ldots,n \}$ for positive integers $n$. These are the prototypes of finite sets.