David, I think your issue over calling the smaller sensor a “1-inch” sensor is down to expecting it to have a 1-inch diagonal. This is not correct, there is a 1-inch dimension in a 1-inch sensor, it has to fit inside a 1-inch diameter of a cathode ray tube.
You need to understand where the term 1-inch and 2/3rds used to describe a sensor size of TV cameras came from. The terms 1-inch, 2/3rds, and 1/2-inch sensors goes back to early days of TV production, when vacuum tubes were used in the camera to capture the image (not CCDs or current single CMOS digital sensors), and the measurement refers to the size of the diameter of the video tube used, and the image sensor size area, inside this cathode ray tube.
A 1-inch tube was one inch in diameter, but the image area was a 4:3 rectangle inside the 1-inch round tube , and this image area was smaller than 1-inch to fit inside the one inch tube. Same goes for a 2/3rds sensor, it is the size of the rectangle drawn inside a 2/3rds of an inch diameter vacuum tube. Original aspect ratio of TV production was 4:3, cameras used tubes for capturing the image, and the terminology used today, is based on this early technology.
So yes, a 1-inch sensor is not one inch in any measurement, but smaller, so it could fit inside a 1-inch vacuum tube sensor area (not quite edge to edge), and it is the circle size of the tube, not actual sensor size, we use to describe their size. Since these tubes only were 1-inch or smaller, this term is only used for these smaller sensor sizes. Four Thirds and Micro Four Thirds sensors are based on a still photography digital camera standard, and both use the same size sensor. Larger sensors are described by the movie film format they are closest to in size, and are not necessarily the same size as their film counterpart. So it all comes down to the terminology we use to describe things.
Also the image from a lens is a circle, and it has to be large enough to cover the sensor corner to corner, so we are back agin to putting our rectangle image sensors inside a circle…
Cheers