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]
(..) Well, put it this way: If it rains... ...you'll be the first to know. Father and I are in the minister's box... ...by personal invitation of Cornelius Fudge himself. Don't boast, Draco. There's no need with these people. Do enjoy yourself, won't you? While you can. (..)2005 Harry Potter and the Goblet of Fire