Prime elements in a ring (usually an integral domain) are elements which satisfy Euclid's lemma (so p is a prime element if p | a b implies p | a or p | b), so the prime elements in the ring of integers are 2, -2, 3, -3, and so on. But if someone refers to prime numbers, they mean 2, 3, 5, ... - it's the standard definition. In fact, if I meant to include both positive and negative elements in the context of the integers, I would go out of my way to say so to avoid confusion! I think the key is the use of the word "number" as opposed to "element" (say).

I think the terminology is a convenience - so many results pertain to 2, 3, 5, ... that it's easier to let "prime numbers" refer to them, rather than to have to qualify everything by saying "positive primes" or taking absolute values.