To start with an opinion, I think that this classification exercise is kind of silly. The student is being asked to put functions into some categories without having a clear idea about what those categories mean or are used for. We introduce definitions and categorizations in order to help us understand abstract ideas. A definition without the underlying motivation is quite hard to grasp. As such, this is an exercise in rote regurgitation—I fail to see its value. :\
That being said, I would argue that there is (up to scaling by a constant) only one exponential function:
$$ x \mapsto \mathrm{e}^{x}. $$
This function is the unique solution to the initial value problem
$$\begin{cases}
u' = u \\
u(0) = 1.
\end{cases}$$
The defining characteristic of the natural exponential function is that it is its own derivative. More generally, we have
$$ b^x = \mathrm{e}^{\log(b) x}
\implies \frac{\mathrm{d}}{\mathrm{d}x} b^x = \log(b) \mathrm{e}^{\log(b) x} = \log(b) b^x. $$
Thus a function $x \mapsto b^x$ has the property that it is proportional to its own derivative. This is what I take to be the defining characteristic of an exponential function. That is, the rate at which an exponential function changes is equal to (or, at least, proportional to) the value of the function.
Any function of the form
$$ x \mapsto a \mathrm{e}^{bx} \tag{1}$$
has this property. Functions of the form
$$ x \mapsto a \mathrm{e}^{bx} + k \tag{2}$$
do not have this property. Because this property is important, it is reasonable to classify these two types of functions differently. Functions of type (1) are exponential, and functions of type (2) are not.
The problem here is that exponential functions are transcendental. You can't really discuss transcendental functions without relying on concepts from analysis (limits, continuity, differentiability, etc). As such, the most important feature of an exponential function (it is proportional to its own derivative) is inaccessible to a student who has not taken calculus. Of course, this renders the question "is this an exponential function or not?" completely mysterious to a precalculus student.