Here is my situation:
To recap my situation, I paid for a new phone, and I suspect I received a refurbished one.
How should I go about addressing this issue? Should I contact Apple? or Verizon?
I will greatly appreciate any advice.
Just to update for others benefit, the message in my OP was a false alarm. The Apple web page for battery recall is for iPhone 5, not applicable to 5s or other models. Still I wish the Apple page doesn't give out false message because to me no information is better than wrong information.
I bought the phone through Verizon, but I contacted Apple first. A technician named Brandon was very courteous and helpful to diagnose the phone setting. We changed few setting in the phone, mostly notification settings. We will see how the battery performs better.