Right and excellent explanation. I was more disagreeing with your definition than saying it could not be done with "cross products" (although it's still bad terminology, biased to people that already know the concepts). That's why I couldn't say you were wrong. Intuitively, a measure of how parallel two vectors of arbitrary dimensions are, is a better default when talking about orientation of vectors.
You need cross product if you want to know sign. Dot product won't tell you which side of the line you're on, and that sounded like a big part of the algorithm.
