With absolutely no disrespect to anyone, I wouldn't worry about being extremely precise on delivering exactly a yard. There is more than enough info here to get in the ballpark, and that is all anyone in sand/rock sales does. I know I've been shorted many times, and I'll mention it if that is the case. I've also gotten a bit extra, even from the same companies that have shorted me. Most owners of the yards don't want to short anyone but unfortunately, it falls under the "weakest link" theory. The weakest link of any business is usually the lowest paid or most disgruntled employee. If a company has one of these loading, or if the guy has had a bad day, is in a hurry, etc, it can happen.
Just figure it out based on the bucket tag/stamping and have the person loading it understand that it is unacceptable to short people. As long as it is in the ballpark, no one is going to complain. Buying rock or sand by the yard is not an exact science. Most places that do volume sell it by the ton and use drive on scales to maximize efficiency and minimize expenses and losses.
That's my two cents!
"Maintenance is Always Cheaper Than Repairs"