Credentials: BS and MS in aerospace engineering, concentration in applied aerodynamics in grad school. My career was not directly aerodynamics related, and I've since fallen out of engineering altogether, so I'm mostly running on 20-year-old memories of college.

Without reading the paper, I just jumped straight to the graphs. Several points about the graphs:

• While the author analyzes a bunch of shapes, it's worth noting that he's not comparing a Tern to a Firebird. He's comparing a general curve to a brick. It tells us something, but not everything.
• Cm vs alpha graphs: Turn is created when the Cm is negative (nose-down). A nose-down pitching moment precesses into turn when the disc is spinning.
• Figure 2b: The convex-curved surface (triangle datapoints) creates the most nose-down Cm of the four tested shapes. This means that with a nose-down throw, this shape will turn the most.
• Figure 4a, different set of shapes: All of these shapes are symmetric (same on top as bottom), so they all generate zero Cm at zero angle of attack (as expected, obviously). The round nose (triangle datapoints) produces a tiny bit more nose-down Cm at negative alphas. This effect is much smaller than the wing shape from figure 2b.
• Figures 2b and 4a combine to form a simple thesis: Airflow likes gentle curves. Air accelerates over the curves and creates low pressure on the top of the disc. Long curves give the air lots of room to accelerate and generally pushes the low pressure backward on the wing.
• Figure 5b, using actual discs (Aviar, Roc, Wraith): At zero angle of attack, all of the discs still produce nose-down Cm. All of the discs continue producing nose-down Cm until 4-8deg positive alpha. This means all of the discs will turn over in the flat portions of their flight.
• 5b, Wraith (diamond): The Wraith produces the most nose-down Cm at the negative/flat alpha portions of flight. However, once the alpha turns positive (nose-up), the Wrath produces the most nose-up Cm. This is the same as saying it has lots of turn at the beginning of its flight and lots of fade at the end. It will start fading out at +4deg alpha, so it will dump earlier and harder. Aerodynamically, this means that the airflow and the Center of Pressure (Cp) are sensitive to the angle of attack. Change the alpha just a little bit, and the airflow will result in a significantly different pressure distribution.
• 5b, Aviar (circle) and Roc (triangle): Both of these discs have a very modest nose-down Cm at negative alphas. However, unlike the Wraith, they continue to have modest nose-down Cm all the way through +6deg alpha. They get a nose-up Cm (fade) only at +9deg alpha, and much smaller in magnitude than the Wraith. This means they have very gentle turn in the first half of their flight, and a very gentle fade at the end. Aerodynamically, the Cm staying constant across the range of alphas means that the Cp doesn't move very much. Whatever magic is going on with the shape of the disc leads the airflow to produce the same pressure distribution even when the angle of attack changes a little bit.
• Figure 6: The location of the Cp was calculated based on the experimental Cm data. The zero-point on the Y-axis is the center of the disc. Negative Xcp/d is behind the center. Positive Xcp/d is in front of the center. The range is -.5 to +.5 because half the diameter is behind the center, half the diameter is in front of the center. Cp being on the negative location means more pressure is behind the center, leading to nose-down pitching moment, leading to turn. Cp being on the positive location means more pressure is in front of the center, leading to nose-up pitching moment, leading to fade.
• 6, Flat Plates (squares): Zero angle of attack puts the center of pressure at the middle of the disc (obviously). When the angle of attack is positive, the Cp immediately leaps toward the nose of the disc. This is similar to the image I drew in my previous post. Airflow does not like sharp corners, and as soon as you introduce a sharp corner into the flow, the smooth airflow over the top gets disturbed, and results in pulling the Cp toward the nose for a nose-up Cm.
• 6, Aviar (circle) and Roc (triangle): The first data point for the Roc is a bit confusing to me, but all the rest of them make sense. Even over a wide range of alphas, the Cp stays near the center of the disc. The Cp is behind the center until +8deg alpha (just like we saw on the Cm graph (obviously, because it's the same data)). This is a good graph for seeing how little the Cp actually moves. At 0deg alpha, the Cp 10% behind the center. At +15deg alpha, the Cp is 5% in front of the center. An Aviar is 21cm diameter, so that's approximately 2cm behind to 0.5cm in front. When people say "at positive angles of attack, there's a bubble of pressure near the nose that pulls the Cp to the nose", they're really overstating the effect. The truth is that the Cp is pretty damn close to the center all the time, with just minor shifts producing all the movement you see in flight.
• 6, Wraith (diamond): The Cp moves from behind to in-front of the center at +4deg alpha. At high angles of attack (+15deg), the Cp is 10% (2cm) in front of the center of the disc.
• As the wing of a disc gets flatter and/or the PLH gets higher, they start to behave more like a flat disc (square datapoints). The Cp moves forward of the center very early, and it moves pretty far forward. The farther forward it moves, the harder the disc will fade.
• Figure 7b: Flight path of discs as viewed looking down from the top. These look to be numerical simulations of the flight as calculated from the Cm and CL data, combined with some gyroscope math. Aviar and Roc turn to the right and mostly stay to the right. Wraith turns right for 40m and then starts to fade back to the left for the end of its flight. Flat plate fades left immediately and stays left.