These are called the natural numbers, NOT the countable numbers.

This is particularly bad because the first four out of your five categories are, in fact, countable.

This is a pretty glaring error, and I can't find anyone else doing this.

Then the second name is also dodgy from the point of view of mathematics - because integers are also "whole" numbers. I do see some elementary texts using this term too, but it feels very wrong to say that -3 is not a whole number.

As a mathematician, if I were teaching people today, I would call both the first two categories "natural numbers".

It's nearly always obvious from context whether 0 is included, and if not we have two separate symbols, N for the natural numbers without 0, and N₀ for the natural numbers with zero.

But calling the natural numbers "countable numbers" when integers and rationals are also countable is simply wrong.