A menu bar is a bar containing a menu (heh), a status bar is a bar containing a status. The menu bar in MacOS has always been called "menu bar", because it is a menu bar. And according to the System Preferences, it is still called "menu bar".
So, why call it here "status bar"??
Agreed. Status Bar is typically associated with Safari.
And I am unable to locate the option for "Automatically hide and show the menu bar app." in System Preferences/General.
Is Menu Bar App new?