Stability vs. Damping
Roullishollandais - good points
I think it is helpful to differentiate between stability and damping. As noted above, if the airplane has a tendency to return when perturbed, that is an indication of stability. If it does not return but settles to a new steady point, it is neutrally stable. If it diverges it is unstable. If it returns with no or minimal oscillation, that shows both stability and good damping. If it returns, but oscillates a lot before it settles back to the trim condition, it is stable but with low damping. If it returns toward trim, but shoots out the other side and begins an oscillation that grows, it has an initial stabilizing moment, but negative damping (i.e., oscillatory instabilty).
Closed loop short period should be both stable and well damped. Closed loop phugoid should not be unstable. With C* there is no phugoid so it is essentially neutral with regard to speed stability. With C*U the phugoid is stable but the damping does not need to be as high as for short period. The phugoid mode is so low in frequency that the pilot has little trouble adding damping through the pitch controller (column/stick).