I'm not sure what the origin might be for the convention. My guess is that since most applications need to allow for multiple trips around the circle, it's more efficient to use extra terminology to specify the less common cases where only the part of the arc that's counted is the part that lies within a single revolution. Bottom line: most appliations involve multiple revolutions so arc distance is defined accordingly.
There's no doubt that the arc length equation is S= theta * radius in radian units.. I just took the definition of an arc "a piece of a circle" and proceeded from there..I think the terminology for this formula could be improved. Arc length would be very appropriate for a length less-than 2pi however, for distances greater-than 2pi the definition of an arc muddies the water.. Anyway, this is why your the professor and I'm the humbled student..