Stability in control theory and electronics
At first - two basic considerations:


The impulse response is a closed-loop test in the TIME domain (and can give you some rough "impression" regarding the degree of stability);
The BODE diagram is an analysis of the loop gain (loop open) in the FREQUENCY domain (and can give you some figures for phase and/or gain margin).


Hence, at first sight, both test are not related to each other. However, the term "stability" has the same meaning in both cases - and the mathematical tools of the system theory connect both domains to each other.

EDIT (UPDATE): Here is the desired answer to your update: 

Regarding stabiliy there is, In principle, no difference between control systems and electronic (opamp based) applications. The DEFINITION of stability is in the TIME domain (BIBO: bounded input gives bounded output), however, the exact proof of stability properties (expressed in terms of stability margins) is conveniently done in the FREQUENCY domain (loop gain analysis). Note that this is one of the main reasons for introducing the frequency domain and the complex frequency variable s.