Is it still a bribe when the public official initiates it? I hope they all get into trouble but I do wonder, when faced with a corrupt public official who is withholding government-provided services, at some point does that not simply become the necessary payment as opposed to a "bribe"?
It's always a bribe. In places without a fully-functional rule of law, it may also be the necessary payment (i.e., the payment of bribes may be expected to conduct business or get through a checkpoint or whatever), but it's still a bribe.