I'm not fond of Brittanica's definition of corruption [https://www.britannica.com/topic/bribery].
Improper and usually unlawful conduct intended to secure a benefit for oneself or another.
It might as well describe any crime, similar to definition c from earlier. Even still, Britannica memtions gifts, which points to corruption being primarily connected to bribery: In societies with a culture of ritualized gift giving, the line between acceptable and unacceptable gifts is often hard to draw.
I suppose I agree with the lack of formal definition for corruption, but defining "corruption" as simply "evil" makes the word ontologically empty, but adding a corrupting element to it (bribe), makes it more defined.
Corruption is a general term, yes. And bribery is a specific crime, which falls under that unbrella. At least in English.