docs: fix to include correct way to authenticate to cloud run instances (googleapis#896)

Corresponding to
googleapis/mcp-toolbox-sdk-python#313

Author: twishabansal

docs/en/how-to/deploy_toolbox.md

@@ -133,22 +133,16 @@ section.
 
 ## Connecting with Toolbox Client SDK
 
-You can connect to Toolbox Cloud Run instances directly through the SDK
+You can connect to Toolbox Cloud Run instances directly through the SDK.
 
 1. [Set up `Cloud Run Invoker` role
    access](https://cloud.google.com/run/docs/securing/managing-access#service-add-principals)
    to your Cloud Run service.
 
-1. Set up [Application Default
+1. (Only for local runs) Set up [Application Default
    Credentials](https://cloud.google.com/docs/authentication/set-up-adc-local-dev-environment)
    for the principle you set up the `Cloud Run Invoker` role access to.
 
-    {{< notice tip >}}
-  If you're working in some other environment than local, set up [environment
-    specific Default
-    Credentials](https://cloud.google.com/docs/authentication/provide-credentials-adc).
-    {{< /notice >}}
-
 1. Run the following to retrieve a non-deterministic URL for the cloud run service:
 
     ```bash
@@ -160,9 +154,11 @@ You can connect to Toolbox Cloud Run instances directly through the SDK
     ```python
     from toolbox_core import ToolboxClient, auth_methods
 
-    auth_token_provider = auth_methods.aget_google_id_token # can also use sync method
-
     # Replace with the Cloud Run service URL generated in the previous step.
+    URL = "https://cloud-run-url.app"
+
+    auth_token_provider = auth_methods.aget_google_id_token(URL) # can also use sync method
+
     async with ToolboxClient(
         URL,
         client_headers={"Authorization": auth_token_provider},