refactor: inline packages/db and packages/types into api package

Phase 2 extraction: groombook/api from groombook/app monorepo.

Changes:
- Move packages/db content to apps/api/src/db/
- Move packages/types content to apps/api/src/types/
- Inline database schema and migrations into api package
- Update Dockerfile to build single package
- Update CI workflow for single-package structure
- Fix vitest.config.ts aliases

Co-Authored-By: Paperclip <noreply@paperclip.ing>

apps/api/migrations/0017_better_auth_tables.sql

@@ -0,0 +1,49 @@
+-- Better-Auth required tables for session-based authentication
+CREATE TABLE "user" (
+  id TEXT PRIMARY KEY,
+  name TEXT NOT NULL,
+  email TEXT NOT NULL UNIQUE,
+  email_verified BOOLEAN NOT NULL DEFAULT false,
+  image TEXT,
+  created_at TIMESTAMPTZ NOT NULL DEFAULT NOW(),
+  updated_at TIMESTAMPTZ NOT NULL DEFAULT NOW()
+);
+
+CREATE TABLE "session" (
+  id TEXT PRIMARY KEY,
+  expires_at TIMESTAMPTZ NOT NULL,
+  token TEXT NOT NULL UNIQUE,
+  ip_address TEXT,
+  user_agent TEXT,
+  user_id TEXT NOT NULL REFERENCES "user"(id) ON DELETE CASCADE,
+  created_at TIMESTAMPTZ NOT NULL DEFAULT NOW(),
+  updated_at TIMESTAMPTZ NOT NULL DEFAULT NOW()
+);
+
+CREATE TABLE "account" (
+  id TEXT PRIMARY KEY,
+  account_id TEXT NOT NULL,
+  provider_id TEXT NOT NULL,
+  user_id TEXT NOT NULL REFERENCES "user"(id) ON DELETE CASCADE,
+  access_token TEXT,
+  refresh_token TEXT,
+  id_token TEXT,
+  access_token_expires_at TIMESTAMPTZ,
+  refresh_token_expires_at TIMESTAMPTZ,
+  scope TEXT,
+  password TEXT,
+  created_at TIMESTAMPTZ NOT NULL DEFAULT NOW(),
+  updated_at TIMESTAMPTZ NOT NULL DEFAULT NOW()
+);
+
+CREATE TABLE "verification" (
+  id TEXT PRIMARY KEY,
+  identifier TEXT NOT NULL,
+  value TEXT NOT NULL,
+  expires_at TIMESTAMPTZ NOT NULL,
+  created_at TIMESTAMPTZ NOT NULL DEFAULT NOW(),
+  updated_at TIMESTAMPTZ NOT NULL DEFAULT NOW()
+);
+
+-- Link staff records to auth identity
+ALTER TABLE staff ADD COLUMN user_id TEXT REFERENCES "user"(id) ON DELETE SET NULL;