Cathode + Anode + Rechargeable battery
I have asked experts -> I am right. The common convention is to label the physical electrodes as cathode/anode based on discharge even though that will be backwards when the battery is charging. That is a poor, confusing, and illogical naming convention, but it is what it is. 

The more clear and logical naming convention would be to name the physical electrodes either positive or negative since those labels apply in both charge/discharge directions. Some people use this logical naming convention, but the illogical convention seems more prevalent.