How to reactivate a iPhone 5s which was originally activated from another account (Device already removed from previous iCloud)?
A full set of instructions are listed here: http://support.apple.com/kb/PH13695?viewlocale=en_US&amp;locale=en_US

In short:


you will need your friend to activate your phone at the Activation Screen during set up. The place you are stuck on.
After that, reset the device by going to Settings > General > Reset.
It will ask for your friends Apple ID details. Enter them.
Once the phone has been reset, tell your friend to remove it from
their Find my iPhone account.
Restart the device and now finally register it under your own Apple
ID.


Activation is a separate process. The device is still registered under your friend's Apple ID. And you are simply registered to use your Apple ID on the device. This is what you're getting hung up on. Resetting the device will allow you to activate it (register the device) under your own Apple ID. Signing out of your friends Apple ID in Settings does no affect device registration.

And of note: If anyone may be attempting to bypass Activation Lock fraudulently or under false pretences, this cannot be done. Apple binds registration of the device to their servers. What that means is Apple will chase that device whenever it is connected to the Internet till the ends of time. Unless you can hack into Apple's servers, there is absolutely no way to bypass Activation Lock.