fix(docker): Fix LLM API key handling for multi-provider support

Previously, the system incorrectly used OPENAI_API_KEY for all LLM providers
due to a hardcoded api_key_env fallback in config.yml. This caused authentication
errors when using non-OpenAI providers like Gemini.

Changes:
- Remove api_key_env from config.yml to let litellm handle provider-specific env vars
- Simplify get_llm_api_key() to return None, allowing litellm to auto-detect keys
- Update validate_llm_provider() to trust litellm's built-in key detection
- Update documentation to reflect the new automatic key handling

The fix leverages litellm's existing capability to automatically find the correct
environment variable for each provider (OPENAI_API_KEY, GEMINI_API_TOKEN, etc.)
without manual configuration.

ref #1291

deploy/docker/api.py

@@ -96,7 +96,7 @@ async def handle_llm_qa(
         response = perform_completion_with_backoff(
             provider=config["llm"]["provider"],
             prompt_with_variables=prompt,
-            api_token=get_llm_api_key(config)
+            api_token=get_llm_api_key(config)  # Returns None to let litellm handle it
         )
 
         return response.choices[0].message.content
@@ -127,7 +127,7 @@ async def process_llm_extraction(
                 "error": error_msg
             })
             return
-        api_key = get_llm_api_key(config, provider)
+        api_key = get_llm_api_key(config, provider)  # Returns None to let litellm handle it
         llm_strategy = LLMExtractionStrategy(
             llm_config=LLMConfig(
                 provider=provider or config["llm"]["provider"],
@@ -203,7 +203,7 @@ async def handle_markdown_request(
                 FilterType.LLM: LLMContentFilter(
                     llm_config=LLMConfig(
                         provider=provider or config["llm"]["provider"],
-                        api_token=get_llm_api_key(config, provider),
+                        api_token=get_llm_api_key(config, provider),  # Returns None to let litellm handle it
                     ),
                     instruction=query or "Extract main content"
                 )