-20 db/decade Gain Slope requirement for Stability?
It is correct that the stability criterion under these conditions (-20db slope at unity gain) is met - that means: fulfilled. 

However, two important remarks:


In your quote, the "overall gain" is mentioned. This sounds misleading to me. To be correct, this stability criterion applies to the LOOP GAIN of a circuit with feedback only (that means: loop not closed).
The mentioned condition ensures a phase margin of app. 90 deg - which certainly fulfills the criterion. However, even with a phase margin of only 10 deg the circuit (after closing the loop) will be stable. But in many cases this margin is not sufficient (too much ringing in the step response). But it is "stable" from the system theory point of view. For many sysytems a margin of app. 60 deg is desired which gives a "good" step response with a small overshoot only.    


(Further explanation: In most cases a gain drop with 20dB/dec is connected witha phase shift of app. -90 deg. Hence, we have a reserve of additional 90 deg until the "critical" value of -180deg).