At what point is a turbine powered ducted propeller considered a turbofan?
tl;dr

They're pretty much all considered turbofans with a few exceptions from the '70s such as this experimental Britten-Norman Islander. They key difference is a prop can be considered a separate entity (add-on) to the turbine for a turboprop, but a fan is an integrated component for a turbofan.



Slightly longer answer

After a fair bit of thought and research I hope I've managed to come up with a satisfactory answer, though to start with a disclaimer. The aviation industry is far too fond of merging and creating new words to describe new engine architecture. Take Open Rotor vs. PropFan for example.

To get technical about it, you could look at the similar/reverse argument about the difference between an open rotor turbofan and a turboprop, as addressed by the EASA.

If you look at Appendix 1: Open Rotor Definition (Page 86) of that document they outlined the following key differences


  Open rotor module that cannot be distinguished as a separate entity


However the following was the agreed definition, which is still rather ambiguous.


  A Turbine Engine featuring contra-rotating fan stages not enclosed within
  a casing


I think the first one is more key, the prop can be considered as a separate entity, attached to the front of a turbine and powered by a gearbox. A fan is an integral part of the engine and cannot really be considered a separate entity, as it actually forms part of the low pressure shaft and is considered the first stage of the low pressure compressor.