Stability of a two poles negative feedback amplifier
To 1: The term "bounded input" has an extreme case: No external input. This is the case for an oscillating system. Hence, the definion (BIBO) is applicable.

To 2: When a pole pair of a closed-loop system has a negative real part, all possible oscillations will die out (due to the term exp(-sigma*t)). Speaking about oscillations, it is necessary to discriminate between 


continuos (steady-state) oscillations with poles on the Im-axis (real part zero)
rising oscillation amplitudes (poles with positive real part)
decreasing oscillation amplitudes (poles with negative real part)