The difference is that bribery is about the "agent-principal" distinction. If an agent who is supposed to represent the interests of the principal actually does

The difference is that bribery is about the "agent-principal" distinction. If an agent who is supposed to represent the interests of the principal actually does so (e.g. follows the policy when making decisions and the money flows as intended by the principal) that's normal behavior; but if they take money to breach their duty of trust, that's a bribe. Since the agent receiving money for UK visa works for the UK, them doing what UK requires is not a bribe - perhaps you might argue that it's extortion or something else, but not a bribe, because there is no breach of duty.