Rotational equilibrium
Rotational equilibrium is the condition in which the net torque on a system is zero, so its angular velocity stays constant. A system spinning at a steady rate is in rotational equilibrium just as much as one that is not turning at all.
Balanced torques, and nothing more than that. EK 5.5.A.1.ii: rotational equilibrium is a configuration of torques such that the net torque exerted on the system is zero.
Constant angular velocity, not zero angular velocity. EK 5.5.A.1.iii gives the rotational analog of Newton's first law: a system will have a constant angular velocity only if the net torque exerted on it is zero. A merry-go-round turning steadily satisfies that. So does a bridge that never turns. Both are in rotational equilibrium, and reading the term as "not rotating" costs marks.
It is independent of translational equilibrium. EK 5.5.A.1 says a system may exhibit rotational equilibrium without being in translational equilibrium, and the reverse. A ball thrown with no spin accelerates through the air with zero net torque about its center of mass. A rod pushed by two equal and opposite forces at opposite ends has zero net force but a nonzero net torque.
EK 5.5.A.2 is the counterpart: if the torques on a rigid system are not balanced, its angular velocity must be changing.
Which torques enter the sum depends on the axis you pick, and picking it so an unknown force contributes nothing is what makes these problems tractable. That routine is in how to calculate torque; the CED framing is at topic 5.5.