Mo,
Foresters, loggers, and sawyers all use the diameter on the small end of a log as the number that defines a log -- if you talk about a 63" log in our part of the world, people will pretty certainly be either confused or amazed.
As for determining the log size required for a certain timber, the math, as you said, is easy. Figure the length of the diagonal across the end of the timber and add a little for defects. You can use Pythagorus, you can use trig, you can measure it directly using a framing square and a tape or rule,you can draw it out full scale, and for square section timbers you can multiply the side length times 1.41. For an 8x8, I look for a straight 14" log, for a 12x12, a straight 18-19" log, for a 6x8, a straight 12" log if boxed heart. These numbers would be tip diameter inside the bark, not butt diameter.
As you know the percentage of heart to sap varies widely from species to species, (think yellow pine and oak). It also can vary widely within species if the trees grew under very different conditions -- fast vs slow.
With sapwood, the only way to know is to cut the tree down and look at the end. If you do it enough, you'll get a feel for what you're looking for, but there are no rules that would apply across different forest types and regions of the country.
