I was taught that whole numbers is a wider set of numbers than natural numbers. While "counting numbers" is a good name for natural numbers, because they are used for counting, the idea of countable set is all about our ability to count elements of this set. We can of course talk about countable set in terms of finding a bijection between the set and natural numbers, but it doesn't matter really. It is just different words to spell the same idea, while (we cannot deny it) counting elements is easier to a beginner. Maybe counting numbers is easier to grasp then natural, maybe it is just authors of textbook thinks so. I cannot understand why are you so bothered with it.

> Just call them their proper name and re-enforce this.

I'll tell you a story. When I learned math in college I was stuck by a Galois theory, I just was unable to understand it. So I decided to look different authors, to find some other approaches to explaining things. I found at least one textbook which I was forced to read from a very beginning, because it used a very different notation which I couldn't understand at all.

It was not the first time I have meet a new (for me) notations for things I already knew. I spent two more years learning math, and finally I understood one thing: there are no such a thing as "proper name" in a math. If you need to refer to a some thing, you need to define it. Math is all about your ability to name and to define things, to invent a math notation to write this things down, and to stick your mind to live inside universe which was built by your definitions, names and notations. It is doesn't matter then how original your notation is. All that matter is what results you can achieve.

From all the branches of math algebra is the most about different notations for the same thing. Just take a look at multiplicative and additive notations for groups. You can use either with any group, nothing matter just your convenience. If you like to use a plus sign and write "a+b" then use additive notation. It you prefer to use no sign and write "ab" then use multiplicative one. In algebra it is your respected right to use notation you like. But it comes with a cost: you need to respect other's right to use any notation they like.

You can do better in your own textbooks :)