TB,
I think you are a little confused with the concept of the sensitive axis and the spin axis. They will always be at 90 degrees to each other.
Try to imagine this:
A gyro is spinning about an axis that is aligned with the wingtips.
If there were only one degree of freedom which was about the longitudinal axis, this would align the sensitive axis with the fore and aft axis of the aircraft. Think here of a gimbal held in bearings fore and aft.
Any rotation about the sensitive axis would cause the gyro to try and precess about the vertical axis, but there is no freedom to move in this direction. In other words the gimbal would oppose this initial precession because it is fixed in the horizontal plane. By opposing this initial precession it is felt by the gyro as a secondary force being applied, and this force when rotated 90 degrees in the direction of spin will secondarily precess the gyro about its free axis i.e the longitudinal axis.
When the gimbal movement is restrained it is a rate gyro, and if unrestrained it is a rate integrating gyro.
What your notes say is therefore correct. The sensitive axis will sense any rotation about it, which will cause the gyro to precess, and it is the precession which is used as the source of the error signal which is then amplified and used by the AFCS.