The link to your slide show still does not work for me, even on the new page.
However, I looked at the presentation linked with the title "Ryo Watanabe's solution" and although too little information is given to completely evaluate the proposal, two independent measurements are used and [u]are required[/u] for stability: the wheel angle (presumably measured by an encoder) and a rate gyro, to measure the angular velocity of the robot body.
So, in this case it vastly oversimplifies the situation to say that only a rate gyro is required.
One way of looking at Watanabe's solution is that the wheel rotational angle, by virtue of a frictional contact with ground, defines "horizontal", which is all you need to properly integrate the rate gyro output. In fact, Watanabe clearly states on slide #10, (in commenting on the approaches used by others):
Information on wheel’s rotation angle is NOT used ==> Internal Stability is NOT achieved
To further emphasize the problem of using a single gyro for a balancing robot: "falling" to the right and "rising" from the left cannot be distinguished from each other, given just a rate gyro.