I'm still confused about what "probability of precipitation" means
I agree that the PoP = C x A that we see on a lot of websites leaves something to be desired.  It gets across the loose idea that the definition involves the area affected as well as the probability of occurrence, which is fine for most casual readers but can be frustrating for more inquisitive minds.  How much are the confidence and the area estimated from ensemble forecasts versus expert judgement?  How was it done before ensemble NWP started in the 1990s?

Anyway, there’s an interesting survey by Stewart et al (2016) that gives a broader overview of how different meteorologists use the phrase “probability of precipitation”.  They categorise the PoP = C x A usage as following the NWS Operations Manual (1984), which is nicely summarized in a comment by Schaefer and Livingstone (1990).  In short, your interpretation in terms of expectations is correct and, as you mention, they describe it in terms of a sum over a hypothetical network of points rather than a continuous integral:

$\textrm{PoP} = \frac{1}{N} \sum_{i=1}^N E[R_i]$,

where $R_i$ equals 1 if it rains at station $i$ and 0 if it doesn't.  In your Pluieville example, this could be approximated as a town with one rain gauge in the north and one in the south, giving PoP = (1.0 + 0.5)/2 = 0.75.

I don't know for sure, but it wouldn't surprise me if these things are still calculated by subsampling gridded NWP models at actual rain gauge network locations, which would allow for comparable long-term verification of the probabilistic forecasts.

Stewart, A.E., C.A. Williams, M.D. Phan, A.L. Horst, E.D. Knox, and J.A. Knox, 2016: Through the Eyes of the Experts: Meteorologists’ Perceptions of the Probability of Precipitation. Wea. Forecasting, 31, 5–17, doi:10.1175/WAF-D-15-0058.1 

National Weather Service, 1984: Zone and local forecasts. NWS Operations Manual W/OM15.

Schaefer, J.T. and R.L. Livingston, 1990: Operational Implications of the “Probability of Precipitation”. Wea. Forecasting, 5, 354–356, doi:10.1175/1520-0434(1990)005&lt;0354:OIOTOP>2.0.CO;2