fix: repair failing tests and type errors across api and shared packages

[stooit]

Summary

Fixes all failing tests and eliminates all tsc --noEmit type errors across the api and shared packages. bun test && tsc --noEmit is now fully green: 22 pass / 0 fail, tsc exit 0.

Root causes & fixes

• Auth middleware (packages/api/src/middleware/auth.ts) — HTTP method comparison for public routes was case-sensitive, so POST /users was not recognised as public and required a token. Normalised the method comparison and removed dead code around the branch.
• Shared types (packages/shared/src/types.ts) — field-name inconsistency on the User type between packages caused a tsc error at the users route. Made the field name consistent with what the route/tests expect.
• Users route (packages/api/src/routes/users.ts) — missing import(s) (e.g. badRequest) caused a Cannot find name type error / runtime failure. Added the missing import(s).
• Pagination utility (packages/shared/src/utils/pagination.ts) — was a stub throwing not implemented. Implemented paginate() to the test contract: correct page slice, second/partial-last pages, total/totalPages, page/pageSize, and empty data for out-of-range pages.
• tsconfig.json — bun-types was installed but never wired into the compiler types, so process (and bun:test) had no global declarations. Added it so these resolve without adding any dependency.

Verification

• bun test → 22 pass, 0 fail
• npx tsc --noEmit → exit 0

Constraints honoured

• No test files modified.
• No new dependencies added (bun-types was already installed).
• Changes limited to the five source/config files required by the failing tests.