I think that modern definitions include zero as a natural number. But sometimes, expecially in analysis courses, it could be more convenient to exclude it.
Pros of considering $0$ not to be a natural number:
generally speaking $0$ is not natural at all. It is special in so many respects;
people naturally start counting from $1$;
the harmonic sequence $1/n$ is defined for any natural number n;
the $1$st number is $1$;
in making limits, $0$ plays a role which is symmetric to $\infty$, and the latter is not a natural number.
Pros of considering $0$ a natural number:
the starting point for set theory is the emptyset, which can be used to represent $0$ in the construction of natural numbers; the number $n$ can be identified as the set of the first $n$ natural numbers;
computers start counting by $0$ (see the explanation of Dijkstra)
the rests in the integer division by a $n$ are $n$ different numbers starting from $0$ to $n-1$;
it is easier to exclude one defined element if we need naturals without zero; instead it is complicated to define a new element if we don't already have it;
integer, real and complex numbers include zero which seems much more important than $1$ in those sets (those sets are symmetric with respect to $0$);
there is a notion to define sets without $0$ (for example $\mathbb R_0$ or $\mathbb R_*$), or positive numbers ($\mathbb R_+$) but not a clear notion to define a set plus $0$;
the degree of a polynomial can be zero, as can be the order of a derivative;
I have seen children measure things with a ruler by aligning the $1$ mark instead of the $0$ mark. It is difficult to explain them why you have to start from $0$ when they are used to start counting from $1$. The marks in the rule identify the end of the centimeters, not the start, since the first centimeter goes from 0 to 1.
An example where counting from $1$ leads to somewhat wrong names is in the names of intervals between musical notes: the interval between C and F is called a fourth, because there are four notes: C, D, E, F. However the distance between C and F is actually three tones. This has the ugly consequence that a fifth above a fourth (4+3) is an octave (7) not a nineth! On the other hand if you put your first finger on the C note of a piano your fourth finger goes to the F note.
I would say that in the natural language the correspondence between cardinal numbers and ordinal numbers is off by one, thus distinguishing two sets of natural numbers, one starting from 0 and one starting from 1st. The 1st of January was day number $0$ of the new year. And zeroth has no meaning in the natural language...