docs: improve sample code consistency and fix API references

- Change Next Steps links from "Client API/Expect API" to "Client Guide/Expect Guide"
- Replace URL object notation with connection string format in PostgreSQL examples
- Fix toHaveHeadersProperty with RegExp to toHaveHeadersPropertyContaining with string
- Change client initialization from `const` to `await using` for proper resource cleanup

Author: lambdalisue

data/index/user_api.probitas.ts

@@ -10,13 +10,7 @@ export default scenario("User API Integration Test", {
   }))
   .resource("db", () =>
     client.sql.postgres.createPostgresClient({
-      url: {
-        host: "localhost",
-        port: 5432,
-        database: "app",
-        username: "testuser",
-        password: "testpassword",
-      },
+      url: "postgres://testuser:testpassword@localhost:5432/app",
     }))
   .resource("http", () =>
     client.http.createHttpClient({