Apply TypeScript rules for Vue code; use official nuxt/eslint package

[rudokemper]

Goal

Closes #44.

What I changed

• Implemented proper use of the @nuxt/eslint module, since other packages were deprecated or not recommended with Nuxt 3.
• Moved types.ts from server/ to its own root-level types/ directory, for usage across the repo.
• Applied TypeScript rules to all vue script code, in so doing resolving loads of type errors - too many to enumerate, and the expectation is NOT that anyone would review them. Suffice to say that the TypeScript errors have gone away and that all non-trivial (e.g. not merely type related) changes have been tested manually in runtime or with vitest.

Note

Immediately upon merging this, I will add the commit to .git-blame-ignore-revs for readability sake.