about the internal workings of a motion detector

If a motion detector gets triggered when something comes within a certain radius, but then something is further away such as 10 percent or 50 percent outside the radius, is it not noticed at all, or does the motion detector calculate how far away it is and then decide whether or not to hit the trigger?

it is not noticed at all as far as the level to trigger the sensor. the sensor doesn’t decide in any thinking sense, it just reacts.

Motion detectors work by comparing the output of several adjacent thermopiles.
If all the outputs are stable, it assumes no heat source is present. If the output of once cell changes, and then the one adjacent to it changes too, then the sensor registers a “hit.”

This behavior reduces false triggering due to stationary heat sources and flashes of sunlight. The sensitivity can be adjusted to make it more sensitive, at the risk of false triggering.

Distance has little to do with triggering (although a large, distant object will trigger multiple cells, which will reduce sensitivity).