What FAA rules govern tethered, outdoor UAV flight?
The FAA's rulemaking document for UAVs says that tethered UAVs are not considered balloons or kites (thanks to Steve V. for finding it):


  [...] a small unmanned aircraft that uses powered systems
  for actions such as propulsion or steering is not a balloon or kite
  subject to part 101


And:


  [...] the FAA notes that the definition of small UAS in this rule includes
  tethered powered small UAS


So a UAV is not a kite and it is subject to all the general UAV regulations (I think they'll be published as 14 CFR 107 at some point). There are also a couple of FAA sources that say that if a tethered aircraft of any type is capable of free flight then all the usual regulations apply. This is from the interpretation you quoted:


  Since 1957, the FAA has repeatedly affirmed its position that any
  vehicle designed for tethered operations only, and not for "free
  flight" should be considered a kite.In contrast, if an ultralight vehicle is designed to be tethered only
  until it achieves a predetermined altitude and is released for free
  flight, that ultralight is an aircraft regulated under part 103
  during the entire operation.


There's also a very clear statement on tethered balloons:


  Free and Tethered Flight Considerations. If a balloon is considered an aircraft under one circumstance (such as free flight),
  it must still be considered an aircraft under other operating
  circumstances, including tethered flight. Tethered balloons are
  considered aircraft, and must be operated in compliance with all
  construction, certification, airworthiness, registration, and
  operating regulations applicable to aircraft.


That comment about "operating circumstances" seems to be the general FAA position on tethered aircraft.