If I understand your question properly, we have two things to consider.
**Stability**
This is the ability for a system to operate under a variety of conditions without self destructing. There are two categories of interest; 1) the ability for the system to return to equilibrium after an initial displacement away from equilibrium and, 2) the ability for the system to produce a bounded output for a bounded input. There are differences between time-varying and nonlinear systems.
**Regulator**
In this system the reference input is zero (since it is missing). It is desired to keep the output as near to zero as possible in the system.
So, in both cases, we certainly want them to remain stable, but one allows two types of inputs, while the other wants to have a constant zero input. One can ask why they defined two different concepts for this and that is a valid question, but they needed a system that didn't have any inputs as opposed to the general case I suppose.