I am confused about the curve carried by the train track. I am using two references: Mosher's note (via arcs) and Johnson's blog (via bands).
By Mosher's definition, I wonder if or not, in the following picture, the blue arc is carried by the train track determined by the red and green arc (we can make the blue arc arbitrarily close to the train track).
Similarly, by Johnson's definition, I wonder if or not the blue arc is carried by the train track determined by the bands from the red and green arc.
No, the blue arc is not carried by the train track.
In my picture, given a path that is close to the train track, in order to guarantee that path is carried by the train track, the tangent lines of that path must also be close to the tangent lines of the train track. This is true for the red and green arcs so they are carried by the train track, but it this is not true for for the blue arc.
In the Johnson picture, each band is a rectangle with two "sides" and two "ends", the ends being where the green and red arcs cross. One thing you have not depicted, but which should be part of the picture, is a foliation of that band by arcs which are (roughly) parallel to the ends, each such foliation arc crossing the rectangle from one side to the opposite side. In order to guarantee that a path that is contained in the union of bands also be carried by the train track, that path should cross the foliation arcs of the band transversely, including the ends of the band. This is true for the red and green paths hence they are carried by the train track. But this is false for the blue path, which as depicted is clearly tangent to one of the ends.