The percentage isn't the percent of it occurring but the area affected. So 51% of thunderstorms means 51% of that town/county/etc. will experience thunderstorms.
That's not how it works. Probability of precipitation means exactly what it sounds like: the probability that precipitation will fall in a specific forecast area.
Interesting. The way I explained it was how it was explained to me by a group of weather forecasters. For instance, here [0], is an explanation from a random weatherman that is in-line with how I was told.
Here's the definition from the NWS: https://forecast.weather.gov/glossary.php?word=probability%2...