refactor: migrate backend to pnpm monorepo with Hono module architecture

- Replace Bun with pnpm 9 + Turborepo + tsx; apps/api renamed to apps/backend
- Split backend into http/, modules/, lib/ layers mirroring tai-specguard
- Add use-case + Result pattern, Problem Details RFC 7807, basePath /api/v1
- Mount Better Auth at /api/v1/auth, keep session-auth whitelist
- Split Prisma schema into prisma/models/*, generate into generated/
- Rework packages/shared into lib/ + schemas/ with pagination DTOs
- Implement health, groups, students, payments, waitlist modules
- Add vitest suite with prisma mocks (21 tests), biome lint
- Point web client to /api/v1/auth and /api/v1/groups/from-organization

apps/backend/src/http/request-id.ts

@@ -0,0 +1,12 @@
+import type { MiddlewareHandler } from 'hono';
+import { runWithRequestLogContext } from '@/logger';
+import type { BackendEnv } from './env';
+
+export const requestIdMiddleware: MiddlewareHandler<BackendEnv> = async (c, next) => {
+  const requestId = crypto.randomUUID();
+
+  c.set('requestId', requestId);
+  c.header('X-Request-Id', requestId);
+
+  await runWithRequestLogContext(requestId, next);
+};