BIBO Stability depends on poles of transfer function, not eigenvalues of A

If internal stability was state remaining bounded, BIBO refers to "Bounded Input-Bounded Output".
This means that by BIBO stability we understand that for a bounded input our output will always be bounded too.

A SISO LTI system is BIBO Stable if the zero-state output response is bounded for all bounded inputs:

An LTI System is BIBO system if and only if all the poles of the transfer function have strictly negative real parts.