Unfortunately, the words such as uncertainty, resolution, accuracy, etc., are used in everyday language do not mean the same as in physics or in engineering, and because of that these concepts lead to confusion.
Anyhow, there really are two completely different concepts frequently confused, namely accuracy and resolution.
Accuracy is essentially the standard ...
The first is correct and the second is diabolically wrong.
People get the idea that Poisson errors mean you just take the square root of everything. No! The thing you take the square root of has to be an actual number of events. Any scaling factors are applied afterwards.