There is just one operation called exponentiation, and it is defined by a power series.
e\^x = sum(x\^n / n!) for n from 0 to infinity
All of the cases you describe here are encompassed by the power series definition, including exponentiation of matrices and operators. Of course, the power series definition probably isn't productive to teach in algebra, as you generally don't learn about sequences and series until later.
Regardless, it might be better to go the opposite direction: instead of saying that there are a bunch of different things called "exponentiation", prove that any *new definition* of exponentiation you introduce is *equivalent to* the other definitions you've already been using in a more limited domain. In other words, restricting to, say, the natural numbers, each definition you've provided is a different way of thinking about the *same* operation. And these ways of thinking have different uses, especially in their ability to generalize.
This idea of gradually generalizing an intuitive concept to areas that aren't so intuitive is extremely important in math, and it seems like saying "we create an entirely new operation whenever we do this" misses the point a bit. What's important (and, sometimes, surprising!) is not how these things are different, but how they're *related,* how you get from one to another, and how that process can be used to develop ideas in much broader contexts.