> The implied difference is that in bribery you're asking the official to do something that they must not do, to have him act against his employer.
That's not the usual definition of "bribery" (though generally it is true of all bribery, including the type you are trying to distinguish, since taking extra personal pay for service is usually formally prohibited, even in places where it isn't effectively enforced, so even what you try to distinguish as "greasing the wheels" as distinct from "bribery" is asking the official to do something that they must not do, and acting against the employer.)
That's not the usual definition of "bribery" (though generally it is true of all bribery, including the type you are trying to distinguish, since taking extra personal pay for service is usually formally prohibited, even in places where it isn't effectively enforced, so even what you try to distinguish as "greasing the wheels" as distinct from "bribery" is asking the official to do something that they must not do, and acting against the employer.)