I got the Logitech C615 ($70 - $75). It works.

It meets the specs, but I had a little trouble with it. The cord is too short, so the vertical field of view is a bit too small. I had to take my first proctored exam hunched over a little so the proctor could see my hands and my entire face at the same time. I added an extension USB cable and it works great now.

Here's the USB extension I used ($7). https://www.amazon.com/gp/product/B019E0FB4G

And the tripod ($11). https://www.amazon.com/gp/product/B00M78G2VO

I'm sure there are cheaper solutions, but I know this set up works.