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.

.env.example

@@ -1,8 +1,8 @@
 # Log level (TRACE, DEBUG, INFO, WARN, ERROR). Defaults to INFO.
 # LOG_LEVEL=DEBUG
 
-# SQLite database path. Defaults to data/c4.db.
-# DB_PATH=data/c4.db
+# SQLite database path. Defaults to data/games.db.
+# DB_PATH=data/games.db
 
 # Application URL for invite links. Defaults to https://games.adriatica.io.
 # APP_URL=http://localhost:7331
@@ -12,5 +12,5 @@
 
 # Goose CLI migration config (only needed for running goose manually)
 GOOSE_DRIVER=sqlite3
-GOOSE_DBSTRING=data/c4.db?_pragma=foreign_keys(1)&_pragma=journal_mode(WAL)
+GOOSE_DBSTRING=data/games.db?_pragma=foreign_keys(1)&_pragma=journal_mode(WAL)
 GOOSE_MIGRATION_DIR=db/migrations