I might create a login for a porn site so that I can have some favorite videos bookmarked and it can figure out the type of material I like. That doesn't mean I want my history saved locally.
Not contradictory at all. These days private browsing for most people just means (1) don't save the browsing history and (2) log me out of all websites temporarily.
But as the other post notes, it goes further (than, for example, Chrome Incognito) in ways that can break sites. Incognito means exactly what you said, while Safari Private Browsing means more.
Not necessarily.
I might create a login for a porn site so that I can have some favorite videos bookmarked and it can figure out the type of material I like. That doesn't mean I want my history saved locally.