Close inspection of the silver tip at the business end of the charger that contacts with the base of the power port led to the discovery of an instruction that turning that silver tip one way would lock it and a quarter turn the other way would unlock it. I tried to unlock quarter turn and ... voila! The power cord worked properly and charged the Garmin.
I have no good explanation for why that locking feature is there. All I can say is that if your charger starts to misbehave as described here, give it a try. You might be pleasantly surprised.
If you can explain the purpose of this lock feature to me, I'd appreciate knowing. Thanks for the help.