I'm not sure I understand the distinction between a limit cycle and a limit cycle oscillation. However, I can explain the difference between a limit cycle and a linear oscillation. If you have a linear system with poles right smack on the imaginary axis, you have a linear oscillator. Depending on how hard you kick your linear system, you can get oscillations at the pole frequency of ANY amplitude.

A limit cycle requires a nonlinearity in the feedback loop. The nonlinearity drives the amplitude of the oscillation to a fixed value, regardless of where it starts or how hard you kick it. You could have other limit cycles and other unstable conditions (like latching) but a particular limit cycle has a very specific amplitude, as well as frequency.

Nonlinear systems can certainly be stable. There are few truly linear systems in the real world. For example, all amplifiers saturate at some point. But even in a purely linear system, it would be extremely difficult to place the closed loop poles exactly on the imaginary axis; component tolerances and drifts always move the poles around. Thus, when a real world system oscillates, it is always due to closed loop poles in the right half plane, i.e. not exactly on the real axis. I think it is fair to say that the oscillations start out linearly because for small enough signals, everything can be approximated as a linear system.  However, since the small signal (i.e. nearly linear) "oscillation" actually has an amplitude that increases exponentially with time, it does not take nlong for the oscillation to run into some physical limitation, a nonlinear effect that reigns in the amplitude. When that happens, we observe a stable oscillation.

There are several methods for predicting the amplitude and frequency of a limit cycle. One of my favorites is the describing function approach. Most books that cover nonlinear control theory will at least touch on this method. I must confess however that I've never studied a ring oscillator so I do not know if describing functions work well with them.  One thing I still remember from the course I took in nonlinear controls (a long time ago) was that the professor made a point of saying there is no single analytic method that works well for all nonlinear systems.

My point is that all systems are nonlinear for large enough signals. Thus, there are examples of stable and unstable circuits all around us. The low noise amplifiers in your cell phone are stable nonlinear circuits. The power supplies in your computer are stable nonlinear circuits. The VCO inside the PLL is an unstable nonlinear circuit. When a nonlinear circuit, like an amplifier, is stable, the circuit behaves approximately linearly in response to small disturbances. You don't notice its nonlinear behavior because it is stable. Apply improper compensation and the amplifier becomes ustable. In an unstable mode, the amplifier must eventually exhibit nonlinear behavior because the voltage swing will eventually clip.