> A city council member trying to balance a budget isn't engaging in corruption by choosing the option that reduces an expense in the city budget, even if it just places the cost on the people

Sounds like a pretty solid definition of corruption to me. Siphon private individuals' money into a corporation instead of using public money to offer public services.

The problem with using an expansive definition like this is that doing so desensitizes people to the kind of corruption where politicians literally just accept money or valuables in exchange for misusing their power. E.g. "All politicians are corrupt, so who cares if he takes bribes? At least he agrees with me on X/Y/Z."

The key difference is that the politician is not being personally enriched by saving money in the city budget this way.