So if you have 2 more years to be Dr. Hil, what does "Ph. D. Pass with Distinction" mean? I assumed it meant you got your PhD, but I'm missing something, aren't I?
It means that I've shown that I know enough to be allowed to go onto the next step toward getting a Ph. D. The other kind of pass is a Masters Pass, which would mean I'd have to graduate this year with just a masters. These exams test a few different fields of math, and it's supposed to be a "no matter what you concentrate in, you need to know at least a certain amount about other stuff, too," so it's pretty much testing a level of knowledge equivalent to a first-year graduate course.
After passing the qualifying exams (which is what I just did), I've got to spend this semester and probably fall semester of next year preparing for the specialty exam, which is pretty much where I decide (well, I've already decided) what field of math I want to concentrate in, then I study all kinds of different stuff within that field, at a more advanced level than the qualifying exams, and get tested on that. While I'm studying that stuff I'm supposed to be thinking about what in particular I want to focus on for my thesis. Then once I pass the specialty exam, I start working on my thesis. Then once that's written, there's the thesis defense, which is pretty much sheer terror. Then, if the thesis committee decides that my thesis and my defense was good enough, then I get to graduate with the doctorate.