Here's an illustration of what they're describing-
See what I'm referring to by "concave path of the rod tip"?
The "tail" in "tailing loop" refers to the shape of the loop, not what is done with the rod after the forward cast is complete.
Any small movement made with the rod after the stop on the forward cast will not affect how the loop unrolls. A "reach cast" will reposition the line nearer the rod tip, but it will not affect how the loop unrolls. What you originally described is functionally a "reach cast" but the reach is on the vertical plane rather than the traditional horizontal plane.
First cast seen is a reach cast-
YouTube - Fly Casting Instruction :: Roll casts, Curve casts and more! :: Cast that Catch Fish