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 shall inform you immediately. There's some sort of high-toned and fancy to-do up at the fort. How could it be two upstanding gentlemen such as yourselves did not merit an invitation? Someone has to make sure this dock stays off limits. It's a fine goal, to be sure. But it seems to me (..)2003 Pirates of the Caribbean: The Curse of the Black Pearl