1.A prime number is a natural number that has no positive divisors other than 1 and itself.
2. A prime number is a number that has exactly two factors - 1 and itself.
Both statements clearly seem to be accounting for 1. Am I missing something here?
I'm not able to read the article because of a 503 timeout, so I can't see this in context, so the following might be irrelevant or off the mark (sorry).
A more interesting/usual pair of definitions is
1. A non-unit p is prime if whenever p divides ab then p divides a or p divides b.
2. A nonzero number n is composite if there are non-units a and b such that n = ab.
Then there is a short proof that, for nonzero non-units, being prime is equivalent to not being composite. (A unit is a number with a reciprocal in the number system. For the integers, that would be 1 and -1.)
The two definitions capture different (but equivalent) parts of the atomic nature of primes.
Elsewhere in mathematics, there are the concepts of irreducibility (whether an object has a subobject) and indecomposibility (whether an object splits into subobjects), where in general irreducible things are indecomposible. Basically, the fact that long division works implies that indecomposible numbers are irreducible and that irreducible factors don't straddle the decompositions.
Definition 2 says that 1 is not a prime: it does not have exactly two factors, but just one.
That's indeed a special case which can be mentioned in the article.