> Private Browsing simply prevents cookies and nothing else

Not true, at least for Firefox. In Private Browsing mode, Firefox will avoid saving temporary files and other data to disk, keeping everything in RAM instead, which is wiped when Private Browsing mode is exited. This includes history, cookies, and cache data. I believe localStorage behaves differently as well.

Firefox also makes an effort to block tracking scripts/servers in Private Browsing mode, using Disconnect's filter list.

The Private Browsing new tab page explains this all pretty succinctly: https://i.imgur.com/MGp53eP.png

You're right there. In this whitepaper we had different questions for each of the browsers to make sure this distinction was noted.

However, users still in general think that private browsing modes do things like prevent their IP address from being seen, or prevent a search engine from recording their searches. So there is still a gap in their expectations.

> I believe localStorage behaves differently as well.

My recollection is that the browser claims to support localStorage, but attempts to use it fail. I don't know why it couldn't just be backed by non-persistent storage.

I'm a full stack web developer. I'm well aware of what the HTTP headers and Javascript methods can expose even while in private. I was oversimplifying fingerprinting a bit.