A bribe is any inducement or reward provided by Person A for Person B to act improperly in order to gain an advantage, whether commercial, contractual, regulatory or personal. It may be money, gifts, hospitality or some other benefit or a promise to provide one of those things in the future.
It is still bribery if it happens in a private business relationship not just if public officers or government is involved - and whether the bribe is provided directly or indirectly. It does not matter whether the person receiving the bribe is the same one who will deliver the advantage.
UK legislation counts improper payments for people to do their job properly (known as 'facilitation payments') as bribes. These typically involve a bribe to avoid an application or notice being 'lost' or put to the bottom of the pile.
It is also a specific offence under the Bribery Act to bribe a foreign public official. For these purposes, public official is very broadly defined.