You're asking the right questions, and honestly, you're reading the texts you have very thoroughly, so that's really a good thing!
The publication you found is wrong about this being single-sided.
Because it doesn't cite the sources for its pictures (which is a big bad thing for a "report"-style publication), one can't verify the claims it makes. That sadly disqualifies the publication as whole. We need to treat all it says with solid scepticism!
Let's, however, really look at this thing in detail:
Here, you see the two separate strings of conductors: the "black" ones, and the white ones. They're actually the same string, connected at the right end
My hypothesis of how they are actually built based on this schematic alone:
The solid parts (green and blue) are on top, the dashed parts (orange and black) below:
And yeah, that's really just a "PCB substrate-core" winding! If you just drilled holes in a piece of board where the little round dots at the corners are, you could wind this with wire yourself. It'd be the same principle as if you took a plastic rod, laid it straight left-to-right in front of you, wound 7 windings clockwise (looking into the rod from the right) around the rod (blue and black), while going slowly from left to right, then slipped a plastic tube around the rod and continued in the same rotational sense for 7 windings around that slightly larger tube, going slowly from right to left (green and orange). Finally, you'd heat the whole thing until the plastic gets soft and squash it flat.
Let's see whether the photograph in its low resolution corroborates that assumption:
These small silvery-pitted circles really look like typical vias to me, indicating that the trace continues on the backside. If it didn't these would all be short unconnected traces, not a zigzag pattern in sight!
So, yeah, this is not a single-layered board, as the text claims. This is really just proper windings on a PCB substrate core :)
As to the general question of what constitutes a winding: the only part of the windings that gets voltage induced is the length that they are perpendicular to the magnetic field lines; the direction of induced voltage, as you say, depends on relative direction of winding and field lines. The fact that a coil with many windings around a core experiences stronger induction stems from the fact that these windings are all in the same direction relative to the magnetic flux.
When you just zigzag in the same plane that's parallel to the magnetic flux, the alternating lines are in opposing directions, and this cancel out.
You need to have a volume that you enclose. That's the PCB here.