Upgrade to TypeScript 5.1.6

I got in trouble while working on #1269 (closed) because the Volar extension on VS Code uses Takeover Mode to support TypeScript inside of Vue components. Their version of TypeScript is the latest, 5.1.6, but the frontend was still using an outdated version, so some type checks that worked properly within VS Code failed when running npm start.

Updating to 5.1.6 caused some warnings to appear, so I upgraded @vue/eslint-config-typescript, as well as fork-ts-checker-webpack-plugin, a dependency of @vue/cli-plugin-typescript.