It sounds like we are discussing two very different things.
What Les Davenport designed is a gyroscope that acts like a spinning top. These have been around for centuries.
There are two ways they can operate
The picture looks like a reaction wheel and is sometimes operated as (and referred to as) a momentum wheel, by operating a flywheel at a constant (or near-constant) rotation speed, in order to imbue a satellite streamliner with a large amount of stored angular momentum. Simply a large mass spinning on a fixed bearing.
The other possibility is to suspend the rotating weight in a single axis gimbal, a Control Moment Gyroscope. The most effective CMGs include only a single gimbal. When the gimbal of such a CMG rotates, the change in direction of the rotor's angular momentum represents a torque that reacts onto the body to which the CMG is mounted, e.g. a spacecraft streamliner. The International Space Station has 4.
The Segway contains neither. It has a type of gyroscope that uses an inexpensive vibrating structure gyroscope manufactured with MEMS (micro-electro-mechanical) technology have become widely available. Aircraft gyroscopes used to cost thousands of dollars. Thanks to the Iphone the cost of MEMS gyros has become dirt cheap. You can buy a three axis gyro for $25. Better yet, you can buy a 3-axis accelerometer, 3-axis gyroscope, and 3-axis magnetometer for $30. About half the size of a credit card.
My $600 quadcopter has one. The software allows it to hover hands-off. Better yet, I can program a route on Google Earth and it will take off, accurately fly the route out of range of the transmitter, and return and land. Substitute an IED for the camera and you can see why everyone is worried about drones!