She attended the fund-raiser 'at the invitation of' the committee chairperson. [=she attended because she was invited by the committee chairperson]
You have a 'standing/open invitation' [=you are always welcome; you are always invited] to visit us any time.
The event is 'by invitation only'. [=only people who have been invited to the event can attend it]
(..) I'm sure she'll find those new geodes of yours fascinating. Come on. I am so glad you accepted my invitation. Now, the kids can resolve their differences before Ms. Grunion arrives. You're barking up the wrong tree, mister. In fact... if it weren't for Patty, I would (..)2014 Mr. Peabody & Sherman