Enhancement: Update types for ESLint v9's changes and deprecations

[JoshuaKGoldberg]

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• I have searched for related issues and found none that match my proposal.
• I have searched the current rule list and found no rules that match my proposal.
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Relevant Package

utils

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• I believe my proposal would be useful to the broader TypeScript community (meaning it is not a niche proposal).

Description

Per https://eslint.org/blog/2023/09/preparing-custom-rules-eslint-v9, there are a few API changes happening in ESLint 8. We'll want to mark types as @deprecated and add new ones in response.

Fail

n/a

Pass

n/a

Additional Info

@G-Rath already got us started in #7812 😄 thanks!

Note that this change should be a non-breaking change at runtime. We still need to support ESLint v8 for the time being.