I would assume the answer to my question is yes, but I want to make sure because my book uses both terminologies. Please also indicate where zero falls into the mix.
Here is an excerpt from my book:
The definition of $\Theta(g(n))$ requires that every member $f(n) \in \Theta(g(n))$ be asymptotically non-negative, that is, that $f(n)$ be non-negative whenever n is sufficiently large. (An asymptotically positive function is one that is positive for all sufficiently large $n$.)