Is positive the same as non-negative?

The real numbers can be partitioned into the positive real numbers, the negative real numbers, and zero. A real number is one and only one of those three possibilities. This is called "trichotomy." Non-negative (or, correspondingly, non-positive) means not negative (not positive), so zero or positive (zero or negative).

That is, non-negative includes zero whereas positive does not.

Edit for clarity:

Non-negative means zero or positive.

Non-positive means zero or negative.

That is, non-negative includes zero whereas positive does not and vice versa.

In mathematical English,

• positive is defined to be $> 0$
• negative is defined to be $< 0$

So non-negative means $\ge 0$, not the same as positive.

In mathematical French, it just happens that the word 'positif' is defined to be $\ge 0$, that is, 0 is both 'positif' and 'negatif'.

In other languages...who knows.

If we go by your edits, about the book excerpt, it looks like the book treats non-negative as $\ge 0$, and positive as $\gt 0$.

Also, from the notation it seems like you are talking about functions whose domain is $\mathbb{N}$.

For an example of an asymptotically positive function, consider

$$ f(n) = 1$$

For an example of an asymptotically non-negative function, consider

$$f(n) = \left|\sin\left(\frac{n\pi}{2}\right)\right|$$

For sufficiently large $\displaystyle n$, we have that $\displaystyle f(n) \ge 0$. Note that this function is not asymptotically positive, because it is zero (for even $\displaystyle n$) infinitely often.

Any asymptotically positive function is also asymptotically non-negative, but not vice-versa.

For an example of a function which is neither asymptotically non-negative, nor asymptotically positive,

$$f(n) = \sin\left(\frac{n\pi}{2}\right)$$

This function takes the values $\displaystyle 1,-1 \ \text{and}\ 0$ infinitely often.