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

[stooit]

Summary

Fixes all failing tests and eliminates all tsc --noEmit errors in this multi-package Hono API repo. Bugs spanned both the api and shared packages.

Result: bun test → 22 pass / 0 fail (4 files); bunx tsc --noEmit → 0 errors.

Fixes

• Pagination (packages/shared/src/utils/pagination.ts) — implemented the paginate() stub (was throw new Error("not implemented")) to satisfy the full test contract: page slicing, total, totalPages (0 when empty), page/pageSize fields, and clamping for out-of-range pages.
• Auth middleware (packages/api/src/middleware/auth.ts) — case-sensitivity bug: public-methods list used "post" but HTTP methods compare as uppercase, so POST /users was not treated as public. Changed to "POST".
• Shared types (packages/shared/src/types.ts) — renamed User.userName → username to match the route handlers, DB usage, and test expectations.
• Route handler (packages/api/src/routes/users.ts) — added the missing badRequest import from ../lib/errors that caused a runtime failure on the 400/missing-fields path.
• tsconfig.json — added "types": ["bun-types"] (already a devDependency) so the process global and bun:test module resolve for tsc.

Constraints honoured

• No test files modified.
• No new dependencies added (bun-types was already present).
• Only source/config files required by the tests were changed.

Assumptions

• The username (lowercase) field name is canonical, since the route handlers, DB layer, and tests all use it — only the shared type diverged.
• The tsc errors for bun:test/process were resolved via tsconfig types rather than editing tests, per the "do not modify tests" constraint.

Verification

Reviewed by the review subagent (approved, no issues) and independently confirmed both bun test and bunx tsc --noEmit pass locally.