It's not just the Internet or Weatherby. In 60 years of shooting I have never heard anyone describe group size as "precision". It has always been referred to as "accuracy".Sorry ... they are also technically using the wrong terminology. They may purposely use the wrong term because that's what people are use to seeing, and don't really understand the difference between precision and accuracy. A rifle's MOA performance is only describing how precisely it shoots. The accuracy is dependent on the sights and if they are "calibrated" (ie, "sighted in") correctly to put the bullets where the shooter wants them. Accuracy is basically adjustable, but the precision isn't ... it's part of the design and manufacturing of the rifle ... you get what you get in terms of its precision - its MOA performance.
The rifle's MOA can be determined without the rifle being accurate ... you just need to see the MOA grouping anywhere on the target to see it (your "Precision Benchrest Compititioin" example). After that, if you adjust (calibrate) the sights to put that group on the bullseye, then the rifle is both percise and accurate, as the many examples given show.
The terminology is used wrong all over the internet, and that locks in misconceptions in people's heads on the actual correct terminology. That's why lots of people really don't know the technical difference between the definition of precision and accuracy wrt the scientific measurements, and also with rifle performance in this context.
Next time you're at the rifle range, ask everyone there if they know the difference between their rifle's precision and accuracy. The are technically two totally different things as shown many times in this thread. The same goes for measuring about anything. A digital micrometer (or any other measuring tool, like machines used in UOAs) might be very precise, but it could be very inaccurate.
In fact the only time I have heard the term "precision" used is in reloading. Where it fits because of the reference to precision, and or precise tolerances.