Rename "Private Search" to "Default Private Search" / make it possible to disable some search engines in Private windows?

[ROTFL]

IMO "Private Search" name is misleading because it suggests newbie users that all searches in private window will be handled only by this search engine.

Which is not exactly true because using search shortcuts in address bar (by purpose or mistake) can cause using a different engine (like Google).

Maybe it's a good idea to add checkbox for each engine "This search engine will be disabled in Private windows"?

[pauloaguia]

I think it's misleading because it might make people think that ticking that box is enough to make the search private. Which is obviously not the case.
So I also agree that the proposed change is a lot less ambiguous.