:::::::I certainly agree that what's going on in the lamp is complicated.  But from what I've read I think a lot of the complexity can be encapsulated in the IV curve using concepts of  

 :::::::There are probably some oscillators and closed-loop control systems that cannot be explained by negative resistance (NR), as you say, but most relaxation oscillators can.  The tunnel diode has N-type negative resistance, while the neon bulb is supposed to have S-type (V is the independent variable in IV curves). In a relaxation oscillator, the negative resistance segment of the IV curve is unstable, so negative resistance cannot be observed directly.  In a relaxation oscillation (limit cycle) the state of the system follows the IV curve, which represents stable states of the NR device, until it reaches the negative resistance section, then "jumps" quickly to the other branch of the IV curve, through unstable states of the device.    

 :::::::N-type (voltage controlled) NR devices like the tunnel diode have hysteresis when driven by current sources, but are stable when driven by voltage sources.   So the whole IV curve including the negative resistance region can be graphed; there's no doubt they have NR.  S-type (current controlled) devices are likewise supposed to be stable when driven by current sources, but I don't know if this is true for the neon bulb.  

 :::::::Hysteresis is evidence of "state", but in relaxation oscillators that includes the state of the energy storage devices in the external circuit.  However, many of the IV curves I've seen for gas discharges show hysteresis in the breakdown region, which would be caused by internal state as you say.   Once breakdown has occurred and the voltage has dropped to the maintaining voltage, the ionization and low voltage can be maintained at currents below the current at which breakdown occurred.  If I understand it, that's because the high electric field in the cathode fall makes it an efficient producer of ions.  But  as I say, S-type NRs are supposed to be stable when driven by current sources.  Do you know if the neon lamp has hysteresis regardless of the driving resistance?  If it is impossible to stabilize the lamp in the breakdown current range at a voltage between the breakdown and maintaining voltage with any external circuit, then I'd agree with you.  The bulb acts like your relay example, switching between two ionization states, and maybe "negative resistance" is not a good term.  But just because NR cannot be observed in the device with one particular external circuit, does not mean it doesn't have it.