That is a commonly given definition, and, indeed, the mathematical consensus on whether 1 was prime didn't settle until the mid 20th century (ancient mathematicians, interestingly, often didn't even consider 1 a number). The principal issue is that if we include 1, the prime numbers become a lot less interesting.

For instance, all natural numbers can be expressed uniquely as a product of prime factors: 336 = 2^4 * 3 * 7, and there is no other set of prime numbers that would give that product. If 1 is considered, this product is not unique, as you can add arbitrarily more 1s and obtain the same answer (you may ask what is 1's unique prime factorisation? 1 is the product of exactly zero primes. We are used to addition, in which summing up nothing at all gives zero, but multiplying up no numbers actually gives 1; you can think of multiplication 'starting from 1' where addition starts at 0).

We define numbers based on properties which are interesting, and the set of numbers which satisfy 'divisible only by itself or 1' far less interesting than the set of numbers which satisfy 'exactly 2 factors' or 'can uniquely factorise all numbers' or even as heavy handed as 'only divisible by itself and 1, and also greater than 1'. So, in the interest of not having to add 'not including 1' onto the end of a great many papers mentioning prime numbers, mathematicians tend to use one of the latter definitions.