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

[stooit]

Summary

Fixes all failing tests and TypeScript errors in the multi-package repo. bun test now reports 22 pass / 0 fail and bunx tsc --noEmit exits clean. No test files were modified and no dependencies were added.

Fixes

• Auth middleware (packages/api/src/middleware/auth.ts): fixed an HTTP-method case-sensitivity bug in the public allow-list (["GET", "post"] → ["GET", "POST"] with .toUpperCase() normalisation) so unauthenticated public GET/POST routes are matched correctly. DELETE/PUT/PATCH still correctly require a token.
• Shared types (packages/shared/src/types.ts): aligned the User field to username (was userName) so the shared type, both API route handlers, and the tests agree. Renaming the type — rather than the routes — was the correct direction, since the tests (which must not change) use username.
• Users route (packages/api/src/routes/users.ts): restored the missing badRequest import from lib/errors.ts (already exported and already imported in the sibling posts.ts), so invalid payloads return 400 instead of a 500 ReferenceError.
• Pagination (packages/shared/src/utils/pagination.ts): implemented the previously-stubbed paginate() to satisfy the shared test contract — correct page slicing, page/pageSize/total/totalPages, out-of-range → empty data, and empty-input handling. Verified no off-by-one.
• tsconfig (tsconfig.json): added "types": ["bun-types"] (already present in devDependencies + node_modules) so process and bun:test resolve. Wires up an existing dependency; the alternative (@types/node) would have been a new dependency, which was disallowed.

Verification

• bun test -> 22 pass / 0 fail
• bunx tsc --noEmit -> clean (exit 0)
• No test files modified; no dependencies added.

Assumptions

• The username/userName inconsistency was resolved toward username because the (unmodifiable) tests assert on username.
• bun-types was preferred over @types/node to honour the no-new-dependencies constraint.

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