I like to treat the naturals as starting at zero, but not for any mathematical reason. I just think that since it's easy to write ℤ+, it's more useful for ℕ to refer to something slightly different than it would be for ℕ to be a synonym for ℤ+ while anyone wanting to include zero in their set was forced to spell out "nonnegative integers".

A dichotomy between "ℤ+" on the one hand versus "ℕ" on the other is just plain more convenient than a dichotomy between "ℤ+" and "ℕ" on the one hand versus "ℤ\ℤ-" on the other.

It is because you mean two different things by "natural numbers", even though the same words are used for things that are not the same things.

I prefer to start by zero; I think it is more useful in general, and makes more sense mathematically for many (although not all) purposes, and systems that you will find objects and operations that have these properties too.