What is the definition of 'small signal bandwidth' and 'large signal bandwidth'? If I google them, I only encounter them in an electrical context (transistors etc.). However, I am looking for a general definition (for any time signal).
In communications, whether a signal is narrowband (has a small bandwidth) or wideband (has a large bandwidth) is relative to the channel's impulse response.
If the channel behaves as a flat filter (constant gain) over the signal's bandwidth, then the signal is narrowband. If the channel introduces significant distortion (that is, if different signal frequencies see different gain) then it is said that the signal is wideband.
Do you perhaps mean small signal model? That is often used in the context of analysing the performance of transistor circuits. The transistors themselves are nonlinear devices, but they can be analysed as linear devices if you choose their operating regime appropriately.
Conversely, the large signal model is when the operating regime is such that the nonlinear effects must be taken into account.