What does "Brushless DC is an AC induction motor with built-in speed feedback" mean?
It means the poster doesn't know the difference between an induction machine and a synchronous machine, nor do they know the range of DC brushless motors out there.

Having waxed cynical: the term "Brushless DC motor" is imprecise.  The best-fitting explanation for what people usually mean is that it's a brushless motor that's got a magnetic design that makes it tend to have a trapezoidal back-EMF profile.  This, in turn, means that the drive electronics can be simplified, in the sense that the excitation doesn't need to be sinusoidal.

Induction Machine: A motor (or generator) that works by a rotating magnetic field that induces current in the rotor, which in turn generates a magnetic field that in turn generates torque.  (Alternately, you can think of the rotor as a magnetic brake trying to stop the rotating field, which generates torque, but that's kind of a wacky way to think of it).

Synchronous Machine: A motor (or generator) that works by a rotating magnetic field, and a fixed field in the rotor.  This forces the rotor to turn at some exact fraction of the excitation frequency of the field (hence, synchronous).

A brushless motor is constructed to be a synchronous machine, but the rotating field is timed to match the position of the rotor to generate torque.

Contrast these with a conventional DC brushed motor where the rotor magnetic field is switched, and there's an unchanging field that's fixed to the frame.  The rotor switching is called "commutating"; that term is carried out to the method used to make brushless motors ("AC" or "DC", although as I mention above the terms are vague) stay synchronized.