From the Naval Air Training Command again.
In part B of Figure 4-1 the maximum displacements are shown to occur 90 degrees after the applied force. This delay in maximum displacement is called phase lag, and is a property of all rotating systems acted on by a periodic force. For a system that is hinged at the axis of rotation the phase lag is 90 degrees. An applied force causes maximum displacement 90 degrees later in the cycle, in the direction of rotation. A system that is hinged at some distance from the axis of rotation (like a fully articulated rotor head) has a phase lag of slightly less.
I wonder if reference to the gyroscopic comes from the high lighted portion of text, both gyro and rotor are rotating systems being acted upon by a periodic force.