[smg][ci] Migrate Response API e2e tests to shared infrastructure (#16680)

Co-authored-by: Chang Su <chang.s.su@oracle.com>
Co-authored-by: key4ng <rukeyang@gmail.com>

.github/workflows/pr-test-rust.yml

@@ -182,6 +182,15 @@ jobs:
             setup_oracle: true
             setup_brave: true
             parallel_opts: ""  # Legacy tests, not yet migrated for parallel
+          - name: responses
+            timeout: 45
+            test_dirs: "e2e_test/responses"
+            extra_deps: ""
+            env_vars: "SHOW_WORKER_LOGS=0 SHOW_ROUTER_LOGS=1"
+            reruns: "--reruns 2 --reruns-delay 5"
+            setup_oracle: true
+            setup_brave: true
+            parallel_opts: ""  # Cloud backend tests not compatible with parallel execution
           - name: grpc
             timeout: 32
             test_dirs: "e2e_test/e2e_grpc"