What does this statement about saving battery power on iOS mean?
It is actually neither of the two. It is a third variant, namely that ordinary apps cannot run in the background. So when you're looking at some app, other apps in the background aren't consuming CPU time (which translates into battery power) - so quitting the app won't make it consume less.
In regarding to receiving notifications, which you mentioned, they will be received no matter if the app is quit or not. In some cases more battery power will actually be used here if you quit the app instead of leaving it be.
In general you should not be quitting apps unless you're experiencing some kind of bug or problem with the app where it doesn't work.