The two types of stability and "Why exponential"
Let's consider discrete-time systems. The generalization to continuous-time systems is straightforward.

First note that BIBO stability does not require that the impulse response be square summable. It is necessary and sufficient that the impulse response be absolutely summable:

$$\sum_{n=-\infty}^{\infty}\left|h[n]\right|&lt;\infty\tag{1}$$

In the context of stability of a causal LTI system, an exponentially bounded impulse response must satisfy

$$|h[n]|\le a\rho^{n},\quad n\ge 0\tag{2}$$

for some positive constants $a$ and $\rho$, with $0&lt;\rho&lt;1$.

If a causal LTI system is described by a rational transfer function, it can be shown that condition (1) for BIBO stability is equivalent to the condition that all poles of the transfer function lie inside the unit circle of the $z$-plane. This is equivalent with the system's impulse response being a weighted sum of decaying exponentials, and, consequently, condition (2) is always met by choosing $\rho$ equal to the largest pole radius, and by selecting an appropriate value for $a$.

In sum, condition (2) is a special case of general BIBO stability for LTI systems with rational transfer functions.

Also take a look at this answer, which also mentions non-linear systems.