The policies are indeed confusing. FAA rules prohibit power banks and spare (uninstalled) batteries from checked baggage.

It's a bit of a grey area on jurisdiction because FAA cares about flight safety (fires) whilst TSA is primarily looking for terrorists.

United Airlines, however, prohibits laptops and tablets:

* Remove any lithium batteries from electronic devices stored in checked bags.

* If batteries cannot be removed, these devices must be stowed in cabin bags only.

* Store any spare batteries in cabin bags.