Okay.. so 1 should neither be prime nor composite. Because -
a) 1 cannot be written as a product of two different factors : ruling out 1 to be composite
b) 1 has only 1 positive divisor : ruling it out to be prime
That's indeed a special case which can be mentioned in the article.
I would simply not define "prime" or "composite" for 1, yes. If you check abstract algebra books (or wikipedia [1]), you'll usually find definitions along the lines of "a non-invertible, non-zero element is prime if and only if ...", and the nice thing about this definition is that it is a useful concept in more general structures than just natural numbers, namely (semi)rings.
Definition 2 says that 1 is not a prime: it does not have exactly two factors, but just one.