What makes you think the site is wrong? I did not verify the numbers for all cases as it gave what I was loosely expecting. Do you think my formula is incorrect? This is in context to this very specific application.
My BSE from Department of Mechanical Engineering says it is wrong. There is no formula because there is no correction. The point of application of moments/couples on a beam make no difference. There is no lever arm because the head of the torque wrench is a pivot.
A couple is a pair of equal but opposite forces a distant apart, “torque”. M = F x D
Now consider a beam with two couples applied anywhere you wish. Now calculate the sum of forces times distance from any point on the beam (an end is usually easiest.) Break the couples into two component forces, let the forces contributing clockwise rotation positive, counterclockwise negative.
Now move one of the couples and repeat. The answer will be the same. It does not matter where the couples are applied. It does not matter how long the beam.
Another way saying same thing:
If the distance between the two forces of a couple is D, and one is on the beam at d1, the other at d2, then D = d1 - d2.
Sum of the forces about the end is
M = +F x d1 + -F x d2
M = F( d1 - d2 )
M = FD
... right back where we started, the distance from origin cancels out.
The error commonly propagated wrongly assumes the torque wrench measures force at handle times lever arm to the nut. The spanner extension is not part of the lever.