I've heard that before, but I'm not sure that it is true. I've always understood it as that is the percentage chance of rain at any point in the given area, say a city. Weather.gov seems to support that:
https://www.weather.gov/media/pah/WeatherEducation/pop.pdf
I think the confusion comes from how they arrive at the percentage, which is close to what you were saying but the conclusion people make from it is wrong:
So as they state in the paragraph in the 1st quote, if you see 40% for a city or area then it boils down to a 40% chance of rain at any given point in the total area.