docs: align lifecycle test output

Update the lifecycle sanity check to match the client output shape. Gateway
state changes are reflected in the periodic `client diagnostics:` line, and the
user-facing missing-gateway diagnostic is printed as a warning.

This keeps the manual Windows E2E guide from asking the tester to find exact
standalone lines that the current client does not emit.

Test Plan:
- git diff --check
- git diff --cached --check

Refs: MVP E2E handoff

TESTING.md

@@ -288,8 +288,8 @@ Expected client output:
 
 ```text
 relay event: LAN gateway disconnected (peer ..., Normal)
-gateway connected no
-Waiting for LAN gateway
+client diagnostics: relay reachable yes gateway connected no ...
+Warning: Waiting for LAN gateway
 ```
 
 2. Start the gateway again with the same relay, room, cert, and interface.
@@ -298,11 +298,13 @@ Expected client output:
 
 ```text
 relay event: LAN gateway connected as peer ...
-gateway connected yes
+client diagnostics: relay reachable yes gateway connected yes ...
 Connected to LAN gateway
 ```
 
-After the reconnect, repeat one quick `ping -S <tap-ip> <lan-host-ip>`.
+The diagnostics line is periodic, so it may appear up to one diagnostics
+interval after the lifecycle event. After reconnect, repeat one quick
+`ping -S <tap-ip> <lan-host-ip>`.
 
 ## Useful Log Signals