Increasing the pressure in a tire changes not only its circumferemce, it also changes it “cord” profile, that shape you’d get should you cut across the tire and look at the shape. Just as increasing the pressure in a balloon makes it get rounder, increasing the pressure in a tire causes the “cord profile” to get rounder. That increases the rolling circumference and reduces the width of the tread that’s in contact with the tarmac.
The question brought to bear is that if the rolling circumference (diameter) actually gets smaller with loss of air, why is it that the surface of the tire can still turn along with the wheel just as before. The answer is that it cannot and does not. The difference on a low ttire between the tread circumference and the rolling circumference is absorbed by squirming and scrubbing of the tread on the pavement. That’s why the rolling resistance of a tire with 10 pounds in it is so much higher than the rolling resistance of a tire with 30 pounds in it. It’s the stretching and squirming of the rubber both on the pavement and in the sidewalls that’s eating up that additional energy.
That squirming of the rubber also generates heat and can (and does) cause sudden tire failure. Try an experiment. Take three old spoons (new ones if you live alone). Bend the handle of the first back and forth about 5 degrees 10 times. You may feel some warmth at the bend point. Now take the second spoon and bend the handle back & forth 45 degrees 10 times. The bend point will be getting hot. Now do the same thing with the third spoon, only to 90 degrees. The bend point will get very hot and the handle may fail. Exactly the same process is happening inside your tires. And the more the rubber has to shtretch and flex, the hotter the tire gets. At some point it’ll fail.
As to the Tire Pressure Monitoring Systems that use the wheel speed sensors, the tire with the low pressure has a smaller rolling circumference and turne more rapidly for a given speed. The difference in speed trips the TPMS light.