docs: correct OAuth enum value and split boot-log block by mode

Reviewer found two accuracy issues in the rewritten "Check Deployment
Mode" section:

- The AuthMode.OAUTH_SINGLE_AUDIENCE enum value is `oauth_single`, not
  `oauth_single_audience` (config_validators.py:28). A user grepping
  their container logs would have found nothing.
- The "Configuring MCP server for <mode> mode" line was presented as a
  uniform <mode> substitution, but app.py:1170 hardcodes the literal
  string `OAuth mode` for OAuth, while app.py:1239 uses the enum value
  for the two BasicAuth modes.

Split the boot-time block into per-mode subsections so each one shows
the actual literal text users will see, and add a one-line note
calling out the OAuth string difference.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

docs/running.md

@@ -336,19 +336,29 @@ messages in the container logs:
 **At server boot (all modes):**
 ```
 INFO     ✅ Configuration validated successfully for <mode> mode
-INFO     Configuring MCP server for <mode> mode
 INFO     Health check endpoints enabled: /health/live, /health/ready
 ```
 
 `<mode>` is one of `single_user_basic`, `multi_user_basic`, or
-`oauth_single_audience`, matching the `MCP_DEPLOYMENT_MODE` setting.
+`oauth_single`, matching the `MCP_DEPLOYMENT_MODE` setting.
+
+**Additional BasicAuth-mode messages (at server boot):**
+```
+INFO     Configuring MCP server for <mode> mode
+```
+
+Here `<mode>` is the enum value (`single_user_basic` or `multi_user_basic`).
 
 **Additional OAuth-mode messages (at server boot):**
 ```
+INFO     Configuring MCP server for OAuth mode
 INFO     OAuth client ready: <client-id>...
 INFO     OAuth configuration complete
 ```
 
+Note the OAuth boot line logs the literal string `OAuth mode`, not the enum
+value `oauth_single`.
+
 **Additional single-user BasicAuth messages (per MCP session):**
 
 These fire when the first MCP client connects, not at server boot — if you