docs: fix stream_mode examples for runs stream (#1033) (#1039)

backend/docs/API.md

@@ -92,10 +92,14 @@ Content-Type: application/json
       "is_plan_mode": false
     }
   },
-  "stream_mode": ["values", "messages"]
+  "stream_mode": ["values", "messages-tuple", "custom"]
 }
 ```
 
+**Stream Mode Compatibility:**
+- Use: `values`, `messages-tuple`, `custom`, `updates`, `events`, `debug`, `tasks`, `checkpoints`
+- Do not use: `tools` (deprecated/invalid in current `langgraph-api` and will trigger schema validation errors)
+
 **Configurable Options:**
 - `model_name` (string): Override the default model
 - `thinking_enabled` (boolean): Enable extended thinking for supported models
@@ -555,7 +559,7 @@ async for event in client.runs.stream(
     "lead_agent",
     input={"messages": [{"role": "user", "content": "Hello"}]},
     config={"configurable": {"model_name": "gpt-4"}},
-    stream_mode=["values", "messages"],
+    stream_mode=["values", "messages-tuple", "custom"],
 ):
     print(event)
 ```