refactor: rename remaining c4 references to games

Update binary name, DB path, session cookie, deploy scripts, systemd
service, Docker config, CI workflow, and .dockerignore. Remove stale
Claude command and settings files.

deploy/games.service

@@ -0,0 +1,24 @@
+[Unit]
+Description=Games Lobby
+After=network.target
+
+[Service]
+Type=simple
+User=games
+Group=games
+WorkingDirectory=/opt/games
+ExecStart=/opt/games/games
+Restart=on-failure
+RestartSec=5
+
+Environment=PORT=8080
+
+# Hardening
+NoNewPrivileges=true
+ProtectSystem=strict
+ProtectHome=true
+ReadWritePaths=/opt/games
+PrivateTmp=true
+
+[Install]
+WantedBy=multi-user.target