What are the "whole numbers"?

The Wikipedia page claims that "whole numbers" can refer to the integers, or the nonnegative integers, or the strictly positive integers. A hidden comment gives a few examples of each usage, which I reproduce below:

Whole number as nonnegative integer:

• Bourbaki, N. Elements of Mathematics: Theory of Sets. Paris: Hermann, 1968.
• Halmos, P. R. Naive Set Theory. New York: Springer-Verlag, 1974.
• The Math Forum, in explaining real numbers, describes "whole number" as "0, 1, 2, 3, ...".

Whole number as positive integer:

• The Math Forum, in explaining perfect numbers, describes whole number as "an integer greater than zero".
• Eric W. Weisstein. "Whole Number." From MathWorld—A Wolfram Web Resource. (Weisstein's primary definition is as positive integer. However, he acknowledges other definitions of whole number, and is the source of the reference to Bourbaki and Halmos above.)

Whole number as integer:

• Alan F. Beardon, Professor in Complex Analysis at the University of Cambridge: "of course a whole number can be negative..."
• The American Heritage Dictionary of the English Language, 4th edition, includes all three possibilities as definitions of whole number.
• Webster's Third New International Dictionary (Unabridged) has the following entry: "whole number n : INTEGER".

From Wikipedia, you have that

Whole number is a term with inconsistent definitions by different authors. All distinguish whole numbers from fractions and numbers with fractional parts.

Whole numbers may refer to:

1. Natural numbers in sense (1, 2, 3, ...) — the positive integers
2. Natural numbers in sense (0, 1, 2, 3, ...) — the non-negative integers
3. All integers (..., -3, -2, -1, 0, 1, 2, 3, ...)

So it seems it is rather not good to use such terminology and rather stick to the more descriptive positive/negative/non-negative integers, naturals, etc.

Almost nobody uses the terminology "whole numbers." Usually, people refer to {0, 1, 2, ...} as the natural numbers (and sometimes they don't include 0). In general, whatever text you're using will provide a definition and that will be consistent within the text but not necessarily outside it.

This is the case with many mathematical definition--internal consistency within texts, but no guarantee of universal adoption. Generally, however, the definitions of a given concept is similar enough across texts that the same methods of proof apply regardless.