It can be either a CCS or a CVS. The speed is related to voltage, and the torque is related to current. Forcing constant torque results in constant current. As the load resistance varies, the voltage varies with it. Forcing constant speed results in constant voltage. As the load resistance changes, current varies inversely.
In real world power transmission, constant voltage is the method employed. Insulators lose much less power than conductors. With constant voltage, to turn off a load, say a lamp, we open a switch and block a voltage source resulting in near zero current. The loss is V^2*G, where G is the conductance of the insulation. If constant current was used, to turn a lamp off, we would close a switch across the lamp shorting it resulting in near zero voltage across it. The loss in this off state is I^2*R, where R is the conductor resistance.
The long and short of it is that V^2*G is much lower than I^2*R. So for now, power is distributed in the constant voltage mode. This is achieved by holding a constant speed, rpm, on the generator. A side benefit to constant voltage operation is constant frequency. The steady 50 or 60 Hz frequency can be employed to make synchronous motors run at a fixed predictable speed. If constant current ac generators were employed, the frequency would vary.
Does this help?