The standard definition of a prime that we all learned is: an integer which is divisible only by itself and one. Additionally, we exclude $\pm1$ from being a prime now a days (though this hasn't always been the case) despite the similarities with primes in this defining trait.
Making a special exception for $\pm1$ feels a bit arbitrary, and leads to confusion; at least among lay people. Instead, I propose the following equivalent definition:
An integer p is said to be a prime if and only if p is coprime to all integers except for multiples of p.
This excludes $\pm1$ since $\pm1$ is coprime to all integers, including its multiples. And it of course excludes composites since a composite c will not be coprime to its prime factors. But the prime factors are not a multiple of c, and thus c is not prime. This definition also works for the negative primes, since a prime $p$ is a multiple of $-p$ and visa versa.
Are there any problems with this definition? Are there any better/more useful ones out there? Thoughts?
EDIT:
Two integers x and y are said to be coprime if and only if $\gcd(x, y) = 1$.
An issue with your definition is that it does not exclude $\pm1$ as you suggest. Expanding the case of 1, for example, we get
But 1 is trivially coprime to all integers in the set {}.
I sympathize with your desire to make the definition of prime more natural, but as written this doesn't get the job done.
From your comments below, it appears that you want
Unlike your original definition, this is equivalent to the standard definition of a (rational) prime. But I don't know that it feels less like an exception. But that's aesthetics; if you like this one better, go with it!