Is there a difference between white paper and design doc? My impression is that a white paper is more academic, and so I'm not sure what it ends up looking like.

At one of my jobs, we wrote design docs for most major changes (we had a checklist for deciding if the doc was necessary at all). It was ultimately quite helpful because documentation about most important changes existed, even if the dev didn't write any documentation in code. The technical designs also had a template, so it was a fairly easy path for a developer to know what questions to answer or not in their doc.

Is this the same idea as a white paper?

I have seen google design docs at google, most are trash. Most design docs everywhere are trash. The confuse more than they explain. I don't care if the idea is good, if the design doc stinks, it will kill a good idea.

At google, the templates for my 1 pager used to be two pages long

Why would you want to work at Google in the first place? Gross