Herschel,
I've attached some quick photos of my 12" Lightbridge base, which shows my cutout and the azimuth circle.
The cutout is approx. 9" long as I didn't want to have to get too close at night to see the reading.
The 3 photos with the Velcro show how the circle rotates about the axis bolt. I could spin it 360 deg if I want to as the Lightbridge has a Lazy Susan bearing (similar to what's shown in the link you provided). If you have Teflon bearings, you still centre it on the bolt, and just cut out sections to allow the circle to pass around the Teflon bearings.
I wouldn't have a circle that sits outside the base board. It will get knocked, look tatty very quickly, and also collect dirt and moisture that will go between your boards.
The circle does not have to come all the way out to the edge of the board, but the closer you get allows a smaller cutout to see it. I've seen different cutover styles and sizes. The best one to use is the one that suits your needs and is "easy" to do. As most base boards are just chipboard. Seal it up after cutting to minimise moisture penetration over time.
Good Luck.
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