Your mind is working fine, it's the inch-based measurement system that's dumb.
The 36x24 mm sensor measurement is straightforward; those are the length and width of the sensor in millimeters. The 36x24 mm size happens to be the same size that the image was on 35mm film, so that size is often called "full frame".
It's the inch-based system that's confusing and unintuitive. It goes back to the time that video sensors were housed in glass tubes. The inch measurement is the diameter of the glass tube that the sensor would be housed in, if it were housed in a glass tube, which it isn't anymore.
An approximate conversion is given by Wikipedia:
Sensor formats of digital cameras are mostly expressed in the non-standardized "inch" system as approximately 1.5 times the length of the diagonal of the sensor.
Wikipedia also has a nice table of nominal sensor sizes. It says a 1" sensor is about 12.8x9.6 mm. This is reasonably close to the specification you found for your camera (it's not exactly the same, but the inch-based scheme is approximate, to sort sensors into "classes" of size, not to specify precise dimensions).