Fix user wait window metric to measure actual user impact (#5275)

* Auto-invalidate game cache when flush marker is updated

During warmup, the staging server may cache stale game data that was
loaded before the active server flushed. Instead of exposing an RPC
for cache invalidation (which leaks internal state), the server now
automatically detects when the flush marker is updated and invalidates
any cached games.

Changes:
- GamesManager: Added `invalidateCacheIfFlushMarkerUpdated()` that
  checks the flush marker's modification time and clears the cache
  if it's been updated since the last check
- Called before checking if a game is in cache, so stale data is
  cleared before any attempt to use it
- Removed the InvalidateGameCache RPC (no longer needed)

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

* Fix user wait window metric to measure actual user impact

The max user wait window was measuring from when nginx switching
started (before recreation) instead of when it completed. This
inflated the metric by ~12s because nginx recreation time was included.

Users can only experience a wait AFTER nginx starts routing to the
staging server, so we now measure from nginx_switch_end (after
recreation completes) to flush_end.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

---------

Co-authored-by: Claude Opus 4.5 <noreply@anthropic.com>

scripts/deploy-blue-green.sh

@@ -290,6 +290,8 @@ main() {
         exit 1
     fi
     log_info "[DEPLOY:${deploy_id}] Traffic switched to eagle-${staging} (lazy-loads will wait for flush marker)"
+    local nginx_switch_end
+    nginx_switch_end=$(date +%s)
 
     # Step 5: Stop active instance (blocks until exit, ensuring flush completes)
     # Users may be lazy-loading on staging during this time - they'll wait for the marker.
@@ -343,7 +345,7 @@ main() {
     local deploy_end_time
     deploy_end_time=$(date +%s)
     local total_duration=$((deploy_end_time - deploy_start_time))
-    local user_wait_window=$((flush_end - nginx_switch_start))
+    local user_wait_window=$((flush_end - nginx_switch_end))
 
     log_info ""
     log_info "========================================="