Hi,
I don´t know the exact definitions.
Generally there are "wanted" oscillations and "unwanted" oscillations.
For sure a sine wave oscillator is oscillation, but not considered to be unstable. (some may say this is a controlled unstable condition)
If an OPAMP input is a sinewave and it´s output is a sinewave, too, then it´s hard to say if it is stable or not.
If the opamp input is a square wave, then it is more "visible".
* on a stable OPAMP you may see no, or only a little overshoot at the output, but then soon becoming flat.
* a less stable OPAMP showing ringing at the output. The longer in time the ringing is, the closer you are to an unstable condition.
* a non stable OPAMP either shows heavy ringing (not becoming quiet) or a continous sine oscillation, or a continous square wave (clipped sine) oscillation at the output.
*****
There are many different cases....
Once I built a low power OPAMP circuit as a frontend for a SAR ADC.
The input to the OPAMP was very low frequency, about DC.
But the ADC builds a switched, capacitive load to the OPAMP output. Every time the ADC´s internal S/H circuit switched there was ringing at the OPAMP´s output. Causing a lot of error at the ADC readings.
Klaus