It's the giving a public institution money only if they do a certain thing that can be compared to a bribe, the morality of said "thing" being irrelevant.

I think it boils down to who has the power: if they start a collection for money to expand their building to have more space and you chose to participate then it's not you dictating what they do with the money, as all you did was see a cause that you found worthy and contribute to it - the power was entirelly in their hands since they could've chosen to collect for a different purpose and you were just a passive agent - whilst if you give them money with the proviso that you get to dictate how it gets used, then the power is in your hands not theirs: the former is more akin to charity and the latter to bribing.

That said, "bribe" is indeed an imperfect metaphor.