Your right, an augmented 6th chord is an augmented triad with an added 6th to it, I don't think anybody is disputing that. These classical European chords were given there name for the interval that was constructed within the chromatic voice leading. In modern music the "classical augmented 6th chords" are nothing more then dominant 7ths constructed off the b6 interval of the scale resolving to the V chord ( F Ab7 G7) although this is more common in minor (F- Ab7 G7#5).

This is one of those things they like to drill into you in theory class in college, just like counter point and all the rules for it (no parallel 5th's or 4th's)There really is no relevance for this stuff in Modern music (regardless of style).

Now that we're all clear on this subject (hopefully), I think you and Griphon should apologize to each other -LOL-
*what happened to Griph? Did he get kicked off?