Let’s say a lobbyist for a nonprofit cancer research group has dinner with a representative and discusses reasons for (lobbies) him to vote for increased cancer research at this dinner. Is this bribery? Because it is lobbying. But under your definition of lobbying and bribing being equivalent terms, they are the same! And to call such an action bribery isn’t really making your case - it’s just nuking the utility of both of the words.
This is a problem with modern political discourse - all rhetoric is flattened. No - lobbying is lobbying. Bribing is bribing.