If an FI establishes a banking relationship with an entity (e.g., major pharmacy or convenience store) who has FI branded-ATMs on premise, am I correct in my interpretation that due diligence need not be performed on those ATMs since they are not "private?"

Correct.
That would not be a Private ATM.

The key in my mind is 'who is responsible for replenishing the cash in the ATM?' Your due diligence should focus on your customer if they replenish. If they just rent space then they are not considered the owner of the ATM.

This instance may be a little less fact sensitive than normal, the ATM is branded as belonging to a bank. As such, it's simply not a private ATM and triggers no due diligence responsibility on the part of the convenience store's bank beyond, perhaps, simply noting its presence and its owner.