Yes you control a motor with a dc current but let's imagine in this way what sort of a current motor see. If you put a oscilloscope on A and ground what type of current wave form you will get.Ah.. Now I see your misconception!
No reversing polarity is NOT making that an AC motor.
EVERY simple brushed DC motor is actually reversing the polarity at least twice per revolution.
that's COMMUTATION!
'sid

Motors produce Back EMF, that much you will agree upon.
Now, a DC motor creates a square wave
(I think we can accept that a single phase potential difference is a infinitely wide square wave)
And so does a BLDC motor it produces a square wave,
or well let's say trapezoidal wave that's a better match; still square'ish.
An AC motor on the other hand produces sinusoidal back EMF.
And since motors like to be powered in the same way they produce back emf, that's the optimal way to excite them.
That's what your wikipedia article means with "does not imply sinusodial..."
There are many mechanical as well as electrical differences between the two.
the rotors are different, the way they like to be powered is different..
But most important of all:
An AC motor has all three stator coils ALWAYS ENERGIZED,
which you simply cannot do with a BLDC.
So please, if you build your controller, make sure you build a Brushless DC motor controller;
Do not build an AC motor controller not even an brushless AC motor controller, it will not power your motor.
'sid