Brazillian law, for instance, defines the crimes of passive and active corruption:

  The Penal Code, in Article 317, defines the crime of passive corruption as "soliciting or receiving, for oneself or for others, directly or indirectly, even if outside the function or before assuming it, an undue advantage, or accepting a promise of such an advantage." [0]

  Active corruption, committed by an outsider, who offers or promises an undue advantage, is provided for in Article 333 of the Brazilian Penal Code. [1]

But, granted, revieweing US and UK law, it seems they don't define "corruption" as a crime (albeit some of the act names do mention corruption). So let's fallback onto the dictionary: [2]

  a: dishonest or illegal behavior especially by powerful people (such as government officials or police officers) : depravity

  b: inducement to wrong by improper or unlawful means (such as bribery)

  c: a departure from the original or from what is pure or correct

Both definition a and c are too ample and, as you put it, "a smorgasbord". Definition b, specially when combined with a, describes something pretty specific: inducement of a powerful agent to wrong by improper or unlawful means, such as bribes.

Embezzlement is better typified under theft. Same goes for most of the others: fraud is fraud, blackmail is blackmail. They may acquire a "corrupt" character when they are done in direct exchange of personal material gains. There are discussions about whether insider trading should be illegal.

Generally speaking, corruption is primarily a crime against public administration because it involves the government, which (supposedly) represents the people. Private companies represent themselves, so they get to (more) trivially decide who is on the line or not.

[0] https://en.wikipedia.org/wiki/Passive_corruption

[1] https://en.wikipedia.org/wiki/Active_corruption

[2] https://www.merriam-webster.com/dictionary/corruption