Add tool-level caching to CI lint-and-typecheck

[jakebromberg]

Context

The lint-and-typecheck CI job re-processes every file from scratch on every run.
TypeScript incremental: true is configured but .tsbuildinfo files aren't cached in CI.
ESLint and Prettier have no --cache flags enabled. Adding file-level caching to all
three tools lets subsequent CI runs skip unchanged files.

Changes

1. Move shared package .tsbuildinfo out of dist/

Both shared/database and shared/authentication have outDir: "./dist" in their tsconfigs,
so tsc --noEmit --incremental writes .tsbuildinfo into dist/. But tsup's clean: true
wipes dist/ before every build, destroying the cached build info. Fix: add tsBuildInfoFile
to place it in .cache/ instead.

2. Add --cache flags to ESLint and Prettier scripts

• --cache-strategy content for ESLint because git checkout resets mtimes; content hashing is stable.
• Prettier uses content hashing by default.
• Explicit --cache-location .cache/<tool>/ keeps all caches in one predictable directory.

3. Ignore .cache/ in tooling

Add .cache to .gitignore, .prettierignore, and ESLint's global ignores.

4. Add CI cache step for tool artifacts

Add actions/cache@v4 step to .github/workflows/test.yml with config-aware cache keys
and cross-commit/cross-branch restore fallbacks.

Files modified

File	Change
shared/database/tsconfig.json	Add tsBuildInfoFile
shared/authentication/tsconfig.json	Add tsBuildInfoFile
package.json	Add --cache flags to lint/format scripts
.gitignore	Add .cache
.prettierignore	Add .cache
eslint.config.mjs	Add '**/.cache/' to ignores
.github/workflows/test.yml	Add tool cache step