docs: fix /health → /health/live, refresh stale mode-detection log examples (#766)

Issue #766 reported that the running.md quick-start tells users to
`curl http://localhost:8000/health`, which returns 404 — the server
only registers `/health/live` and `/health/ready` (K8s-style probes).
The same section also listed BasicAuth and OAuth startup log lines
(`BasicAuth mode detected …`, `OAuth mode detected …`) that no longer
exist anywhere in the codebase.

Update running.md and troubleshooting.md to point at the real
endpoints, explain liveness vs readiness, and replace the fictional
log examples with messages the server actually emits today. Also
clarify that the per-session BasicAuth messages only appear after the
first MCP client connects, which is the second symptom the reporter
hit.

Docs-only change; code paths and endpoint surface unchanged.

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

docs/running.md

@@ -313,30 +313,51 @@ MCP clients (like Claude Desktop, LLM IDEs) can connect to your server:
 
 ### Check Server Health
 
+The server exposes two Kubernetes-style probe endpoints:
+
 ```bash
-# Test if server is responding
-curl http://localhost:8000/health
+# Liveness — server process is up (always 200 if running)
+curl http://localhost:8000/health/live
 
-# Expected response: HTTP 200 OK
+# Readiness — server can reach Nextcloud and (if enabled) Qdrant
+curl http://localhost:8000/health/ready
 ```
 
-### Check OAuth Configuration
+`/health/live` returns `200 OK` as long as the process is running. `/health/ready`
+returns `200 OK` with a JSON body describing each dependency check, or `503` with
+the same JSON body listing which checks failed — use the readiness probe when
+troubleshooting connectivity to Nextcloud or Qdrant.
 
-Look for these log messages on startup:
+### Check Deployment Mode
 
-**OAuth mode:**
+The server logs the detected deployment mode on startup. Look for these
+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.
+
+**Additional OAuth-mode messages (at server boot):**
 ```
-INFO     OAuth mode detected (NEXTCLOUD_USERNAME/PASSWORD not set)
-INFO     Configuring MCP server for OAuth mode
-INFO     OIDC discovery successful
 INFO     OAuth client ready: <client-id>...
-INFO     OAuth initialization complete
+INFO     OAuth configuration complete
 ```
 
-**BasicAuth mode:**
+**Additional single-user BasicAuth messages (per MCP session):**
+
+These fire when the first MCP client connects, not at server boot — if you
+have just started the container and no client has connected yet, you will not
+see them in the logs:
 ```
-INFO     BasicAuth mode detected (NEXTCLOUD_USERNAME/PASSWORD set)
-INFO     Initializing Nextcloud client with BasicAuth
+INFO     Starting MCP session in single-user BasicAuth mode
+INFO     Creating shared Nextcloud client with BasicAuth
+INFO     Client initialization complete
 ```
 
 ---
@@ -416,7 +437,7 @@ Common issues:
 
 ### Can't connect to server
 
-1. Verify server is running: `curl http://localhost:8000/health`
+1. Verify server is running: `curl http://localhost:8000/health/live`
 2. Check firewall settings
 3. Verify host binding (use `0.0.0.0` to allow network access)
 4. Check OAuth authentication if enabled

docs/troubleshooting.md

@@ -290,10 +290,10 @@ kill <pid>
 2. **Test connectivity:**
    ```bash
    # Test from same machine
-   curl http://localhost:8000/health
+   curl http://localhost:8000/health/live
 
    # Test from network (if using --host 0.0.0.0)
-   curl http://<server-ip>:8000/health
+   curl http://<server-ip>:8000/health/live
    ```
 
 3. **Check firewall:**