This is an interesting concept. I can't say I ever really made a division when homeschooling, because parents are still the teachers of their children in many ways, even if they're attending a school. With the example you give of your mother being a teacher and having to take on that route with you in school, that makes sense because she's teaching other children as well. Hence the line that needs drawn in order to not be seen as favouring her own child.
I guess if your creating a fairly formal school environment at home, then there would be more of a distinction between the roles of parent and teacher/principal. I wonder now if this is what is referred to as school at home, rather than homeschooling towards the unschooling side. I think we were somewhere in the middle.
So interesting to be reading about the numerous different approaches.