Class A = both transistors are ON all the time.
Class AB = both transistors are ON at idle, then up to a certain output current. When output current is higher than a certain limit, one of the transistors turns off.
Class B = either one transistor or the other is ON, but not both. The transistor that is ON is determined by output current polarity.
Class C = either one transistor is ON, or the other is ON, or both are OFF. ie, during part of the cycle they are both OFF and no output current flows. This can be used if the load is a tuned circuit (which is excited by the fundamental and rejects harmonics) or a motor, a solenoid, etc. Using this mode is intentional, the goal is high efficiency. The term "class C" also applies to a single transistor power amplifier if the transistor conducts for less than half (180°) of the cycle.
Note that class B doesn't actually exist. Since transistors threshold voltages vary with process and temperature, in practice you will never be able to adjust the bias voltages accurately enough to get class B. You will either get class AB if the gate bias is a bit high, and some conduction overlap will occurs, or if the gate bias is a bit low you will get class C when both transistors are off during part of the cycle.
Class C increases crossover distortion massively, so it is not recommended if you care about distortion. However, it has an advantage: no current is wasted keeping both transistors conducting at idle. So if the application doesn't care about crossover distortion it is a good choice.
Additionally when one considers a transistor to be "off" is a bit debatable, is it off when current is 1mA? or 1µA? or the leakage current? Or a negligible current relative to the output current?...
Anyway. The last schematic (without voltage sources to bias the gates) depends on your FET's threshold voltage.
For example, it can be class AB if you use JFETs which are ON when Vgs=0V.
If you use MOSFETs which require Vgs of a few volts (Vgsth) to begin turning on, then at idle both FETs are off, and you need to move the input voltage by at least one Vgsth up or down to turn one FET on. So it is class C.
Adding the gate bias voltage sources (as in the first schematic) would make it class AB once the bias is high enough to turn both transistors on at idle (zero output current).