Let's begin at the very beginning, because you ask a great question and someone else may come along who knows less about this than you do. And most definitely, there are guys here who know way more than I do and they can share their take on this and we can all learn something.
one article describes the idea of "doubling" angles: "The complete cutting angle of the knife (which is rarely referred to and is composed of the sum of both edge angles) is called the included angle. Since most knife blades are ground symmetrically, in most cases, the included angle for a knife is simply twice the edge angle." http://kitchenknifeguru.com/knives/knife-edges-101/
I think I understand the doubling aspect of the knife edge angle; but are we talking about the primary or secondary bevel? In my 4 degree illustration above, is the primary complete cutting angle 8 degrees? If I grind the secondary bevel at 15 degrees would the complete cutting angle be 30 degrees?
This angle is the secondary bevel, i.e. the cutting edge angle. Here's where the confusion sets in. Most makers refer to the cutting edge angle as the secondary bevel, because the primary bevel would be the blade if it was ground to a zero edge. In other words, the very foundation of a knife blade is a simple triangular wedge. So let's assume that. A full-height flat grind. A pure wedge. Therefore, each side of the blade is the primary bevel. But that knife would be a camp chopper. A kitchen knife for instance doesn't need to be nearly that beefy. So the maker thins the blade. But a super thin blade with a zero grind would be a razor blade and the edge wouldn't last a farting spell. To fix this, the maker grinds a secondary bevel which forms the cutting edge angle.
So: angles. People refer to a 12 degree angle. That is per side. They are referring to the angle of the blade to the sharpening stone. 12 degrees per side would be 24 degrees "inclusive" since we have to include the angle on each side of the blade to describe the wedge as a whole.
It gets confusing because guys like Murray Carter refer to cutting edge as the Primary bevel.
I have to admit, I am confused on how to calculate primary and secondary bevel angles.
Let me begin with this: I make slicers. I want a knife that cuts effortlessly. Guys who make camp choppers will give you a very different approach.
In a nutshell, don't overthink it. Let's call the cutting edge the secondary bevel. That's the one that really matters. How acute you can make this angle will depend primarily on how strong the edge needs to be versus how how sharp you want to go. Super sharp or super strong: pick one. Better steels with proper heat treat can steal a little from both. But it's always a compromise. I want my kitchen knives as sharp as they can be and as thin as practical for most jobs. But a crazy thin, super sharp chef knife will get destroyed if you start cutting a bunch of carrots, pineapples, and butternut squash. Luckily, you don't slice those things quickly or paper-thin. A knife for brute work doesn't need to be scary sharp. Sharp enough will do.
In my opinion, the end use of the knife dictates geometry. For example: I've never in my life needed a knife made from 1/4" steel. I carry a saw to cut anything I'd need that kind of blade for. But opinions vary, and big thick heavy blades seem to be in fashion. Horses for courses, and more power to any maker who finds a niche to sell knives to.
My hunting knives begin at 1/8 inch thickness. My chef knives begin at .110 My fillet knives start at .070 but are .040 at the ricasso and taper to .020 at the tip What does that make the angle of the primary bevel? Heck if I know. I just know how thin/flexible I am willing to tolerate to make a knife that is strong (enough) and light weight, which slices like a laser.
