In high school I was taught that natural numbers are 1,2,3,..., absolute integer (or just absolute) numbers are 0,1,2,3,... and relative integer (or just relative or just integer) numbers are ...,-3,-2,-1,0,1,2,3,... I think absolute numbers is a nice name, since they're just the absolute values of the integers!
BTW, I don't think the point about zero fish is that it doesn't occur in everyday life but that is doesn't occur in Nature, nor zero goals do. After all, we're talking about natural numbers. Sure any kind of number might occur in some law of physics, but maybe what I mean is "everyday nature":)
It is amazing (and nice) how such a trivial topic (is it that important how do you call them as long as you keep consistency?) generates so much discussion!
Um abraco a todos,
Joao Caramalho Domingues Dep. de Matematica Univ. do Minho, Braga, Portugal