Wikipedia constructs an integer as the difference of natural numbers denoted by an
oriented pair. Negation is a flip of the orientation. A positive integer is a
difference, but the literature generally says it can be taken as a natural number.
A positive integer has an additive inverse, a natural number does not. How am I to
to understand a positive integer as a natural number?
On
The important thing to understand is that there exists not the natural numbers, in the sense that each natural number is a specific object. The natural numbers can just as well be given as digit strings, as strings repeating one character (|, ||, |||, …), or as a collection of apples. What is a natural number is determined not by the objects, but by the structure put on the objects.
In particular, with natural numbers, you must have a certain minimal element ($0$ or $1$, depending on your definition), for each element an unique next element, addition and multiplication which fulfills certain rules, and you get all of them by just starting at the first and repeatedly going to the next (this is a quite informal description; the more formal description is given by the Peano axioms).
It can be verified that all those conditions are fulfilled for the positive (or non-negative) integers, therefore those integers describe the natural numbers. Or again, more formally: The positive (or non-negative) integers fulfill the Peano axioms if you identify the natural number $n$ with the integer $+n \equiv (n,0)$, the successor with $n+1$, and the addition and multiplication with the corresponding operations on the integers.
About the additive inverse: A positive integer has an additive inverse in the integers, but it does not have an additive inverse in the positive integers. So when identifying the natural numbers with the positive integers, there does not exist a natural number that is the additive inverse of another natural number.
The set of natural numbers is not a group because additive opposites of natural numbers are not natural numbers.
When we extend the natural numbers to the set of integers, then we allow negative integers and the set of integers will be a group.
Then we can consider the set of natural numbers as a subset of the group of integers, thus a positive integer has an additive opposite which is an integer but is not a natural number.
Thus when we say natural numbers do not have additive opposites, we mean the additive opposite is not a natural number.