What does the term 'min' mean in AAA Rechargeable cell current capacity?
It means that the minimum capacity is 630 mAh. 

This in turn, for an ideal battery, means that it will supply 630 mA for 1 h, 63 mA for 10 h, 6.3 mA for 100 h, etc.

Real batteries don't perform perfectly. The mAh capacity will depend on the discharge rate and will decrease with increasing current. You may find that the 630 mAh is for the 10 h discharge rate and that at 630 mA the discharge time is &lt; 50 minutes. You will also find that output voltage will drop at higher discharge rates due to internal resistance of the cell.

The test conditions that yielded the 630 mAh capacity should be given on the battery datasheet (which Amazon probably won't supply) along with internal resistance.