Pitch diameter is correct, I parsed that phrase when I made it for clarity. Conceptually, it does not matter much.
I might have fat fingered my math. I only check my work when I'm being paid or working on my own projects.
9mm diameter is 28.26mm circumference. What are the odds that the pitch diameter is... um... roughly 25.4mm. Probably coincides with the same error in a 6mm lead screw pitch with thousandth inch handwheels, conversions being constant and all.
If I have a 4 TPI lead screw with an 8 tooth, 2" (pitch circumference) wheel on it, the dial turn twice in two inches of linear travel. You have all the constant values, you have the relationship, you can solve the conversion factor for metric. Or not, it doesn't matter when you start all threading at "1" on the dial. It's looking for other numbers and dial positions that require the measuring systems to match. Otherwise, it's proportions, relationships, and conversions, but no black magic involved.
It's funny how the concept of conversion is simple to me, but the idea of using any number other than "1" on the thread dial isn't appealing to me. It seems the shoe is on the opposite foot for you.