For others who don't like the native PDF reader or desktop apps, check out drawboard!
I'm a student and use it primarily for taking notes in textbooks and lecture powerpoints, but it serves nicely as a general reader as well.
Just a note about Drawboard: It should have come with your surface so you don't have to pay the (admittedly very reasonable) price in the windows store. If you can't find it on your device or in the purchased section of the windows store, contact Drawboard. I purchased a surface 3 (not pro) and it didn't come with it, even though it was supposed to. I contacted them, explained the problem, and after a day or two, they sent me a code to redeem in the store for Drawboard. Excellent software. Edit: This link here is the evidence
Drawboard PDF is your solution. It's on sale right now for $6 and does exactly what you're trying to do in OneNote, which unfortunately doesn't have a robust pdf manipulation system.
Free full version? And source?
EDIT: http://www.drawboard.com/documentation/151007_SurfacePro4_SurfaceBook_Preinstallation.pdf
Does anyone know it remains after reformat? I would assume that since it comes pre-loaded, it's not actually linked to the user's MSFT account.
I use Drawboard PDF for that, it's available for 10$ in the Windows Store (so it needs Windows 8 or higher). It's not without flaws, but it works well enough. As for free alternatives, Xournal is capable of PDF annotation aswell. I haven't spent much time with it, but it appeard to do its job well enough when I tried it.