Properly adjusted refers to setting the proper preload of the bearings during final adjustment. The headset should not bind nor should it have any play. While way too tight and way too loose are easy to detect usually, slightly too tight and slightly too loose can be hard to tell without experience servicing headsets and can cause damage.

Your explanation is not bad but it is hard to tell without seeing the bearing cages and headset first hand or in pictures to know if you inserted one or both of the cages upside down or not.

And yes, all parts should have been degreased and wiped clean and dry, prior to adding fresh grease and reassembly. The old grease is a perfect medium to hide gritty dirt and other contaminates and makes it impossible to adequately inspect the balls for any pitting/damage.

-j