Before beginning my response, I think it important to understand the history of museums, how they came to be, for what purpose and for whom? I rely on my early childhood experience with museums. It was a special and "exotic" fieldtrip of sorts, a place to see prized possessions from places in the world I may not get to. It was a chance to see things of great value that I might never acquire. It was a chance to have a little "power" over situations out of my control, logistically or financially. I do not know the history of first museums, but it seems a very Western concept.
How do museums influence what visitors believe about Africa?
Museums control what visitors can learn about Africa. Just having items in a museum connotates they are "exotic" or other-worldly. I appreciate exhibits that encompass more complete stories behind their display. And every time a museum announces an exhibit on Africa, we are re-cementing that generalization that Africa is one place instead of 53 different countries, each with their own stories, culture and histories.
What is the difference between repatriation, restitution, and reparation—and why do those distinctions matter?
I once heard a historian from the Mohawk nation describe repatriation in this analogy: "My grandpa stole your grandpa's car and I'd like to return it." Continuing in this analogy, restitution would be something like, " My grandpa store your grandpa's car and I'd like to return it, restoring it to the exact shape it was in when it was stolen." Reparition would be, " I'd like to return the stolen car and make up for the difficulty it put your grandpa in at the time, the hardship he faced because he no longer had the car, how it affected his family and livelihood, to put right the frustration and even anger he may have had, or the difficulty he had in reporting the theft and any harm to his reputation, and any lasting affects on his children and grandchildren."
In this simple analogy reparation is the most comprehensive and difficult to complete.
Who should have the authority to decide the future of African cultural heritage held in museums abroad?
As others have mentioned, I feel a council of some kind should be the decision makers on first, how items are obtained, what objects are obtained (respecting cultural values; for example, for some cultures displaying items that represent themes of the after life may be disrespectful), and where they are displayed. Then steps can be taken to determine disagreements on objects museums currently have. It seems that if a an object's country of origin has any hesitations, the process stops there and items are returned. If a country of origin wants to share items for an exhibit or is agreement for foreign museums to house their items, that could be an option.
This underlying thread of power runs throughout this discussion. A balance needs to be found between exhibiting objects, the stories behind the objects and a more equal distribution of power on deciding what objects, where displayed and what stories are told.