NOAA actually has a succinct explanation for part of this: https://www.weather.gov/ffc/popThe short of it is this: there is a X percent chance that rain will occur at any given point in the area over the next hour, independent of the other hourly predictions. This extrapolates to the daily forecast as well, which does have hourly bounds (though I forget what they are).

For a while the local meteorologist changed it from 30% chance of rain to 30% *coverage* of rain, meaning for the viewing area 30% of it was getting rain. I assume that's still what they use even though they switched back to "chance."

Although this was in Mobile, Alabama, and a 30% coverage of rain meant that you were gonna get wet.

Or, at least, 30% of you would. But yeah, the way the husband of a meteorologist I used to work with explained it, "30% chance of rain" means, roughly, "It's going to rain somewhere. It's a 30% chance it'll be on YOU" (Though that is just how he put it. Technically, "30% chance" could mean "It's definitely going to rain, and 30% of the area will get it" or equally "It's only 1-in-3 that it will rain, but if it does, it will rain EVERYWHERE" But these two possibilities come out to the same thing assuming you are a stationary object within the area which is small enough that you can not meaningfully be "half-rained on"

12PM can't be 12:00 exactly, because then it wouldn't be PM, it'd just be M