Linear system stability criteria
Only the first criterion is correct when referring to input-output stability. The second criterion is just a way to compute the poles, as mentioned in a comment. All realizable continuous-time systems using lumped elements have poles, which doesn't contradict the stability requirement.
Poles in the right half-plane (RHP) make a causal system unstable. Note that if we let go of the causality requirement, RHP poles don't cause instability but they make the system non-causal.
Note that poles on the imaginary axis cause transients that don't die out, which is undesirable in almost all practical applications. So in practice you want all poles to lie on the open left half-plane, i.e., $\textrm{Re}\{s_{\infty,k}\}&lt;0$, where $s_{\infty,k}$ is a pole of the system.