Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?
It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers.
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There are the two definitions, as you say. However the set of strictly positive numbers being the natural numbers is actually the older definition. Inclusion of $0$ in the natural numbers is a definition for them that first occurred in the 19th century.
The Peano Axioms for natural numbers take $0$ to be one though, so if you are working with these axioms (and a lot of natural number theory does) then you take $0$ to be a natural number.
On
There is no "official rule", it depends from what you want to do with natural numbers. Originally they started from $1$ because $0$ was not given the status of number.
Nowadays if you see $\mathbb{N}^+$ you may be assured we are talking about numbers from $1$ above; $\mathbb{N}$ is usually for numbers from $0$ above.
[EDIT: the original definitions of Peano axioms, as found in Arithmetices principia: nova methodo, may be found at https://archive.org/details/arithmeticespri00peangoog : look at it. 
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I remember all of my courses at University using only positive integers (not including $0$) for the Natural Numbers. It's possible that they had come to an agreement amongst the Maths Faculty, but during at least two courses we generated the set of natural numbers in ways that wouldn't make sense if $0$ was included.
One involved the cardinality of Sets of Sets, the other defined the natural numbers in terms of the number $1$ and addition only ($0$ and Negative Integers come into the picture later when you define an inverse to addition).
As a result when teaching the difference between Integers and Natural Numbers I always define $0$ as an integer that isn't a Natural Number.
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Peano used 1 as the first natural number in his arithmetic theory. Geometric: If a line is breadthless with infinite points, then the first line you can create is a 1. Written 1 if it were a map you could scale it. This 1 can then be used as a metric to index a length of each successive natural number. On a Cartesian Graph 0 is used to define one end point of the 1, but without the second end point at 1 there is no length, and no number according to some. Also consider, if you were creating a set of lines each representing a natural number, what would 0 look like? There would be no length to 0, and hence it is not a member of the set of lines.
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The Peano-Dedekind axioms (as used in proving propositions by use of the Principle of Mathematical Induction) define the $\mathbb{N}$ as either $\mathbb{N}$ = $\mathbb{Z^+} \cup \text{0} = \text{{0, 1, 2, ...}}$ or $\mathbb{N} = \mathbb{Z^+} = \text{{1, 2, 3, ...}}$, that is, it depends on the context (usually this "context" may be seen from the given proposition to be proved, at least in the case of using PMI).
Simple answer: sometimes yes, sometimes no, it's usually stated (or implied by notation). From the Wikipedia article:
Saying that, more often than not I've seen the natural numbers only representing the 'counting numbers' (i.e. excluding zero). This was the traditional historical definition, and makes more sense to me. Zero is in many ways the 'odd one out' - indeed, historically it was not discovered (described?) until some time after the natural numbers.