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.

Taskfile.yml

@@ -27,9 +27,9 @@ tasks:
       - "assets/css/output.css"
 
   build:
-    desc: Production build to bin/c4
+    desc: Production build to bin/games
     cmds:
-      - go build -o bin/c4 .
+      - go build -o bin/games .
     deps:
       - build:templ
       - build:styles
@@ -49,8 +49,8 @@ tasks:
     cmds:
       - |
         go tool air  \
-         -build.cmd "go build -tags=dev -o tmp/bin/c4 ." \
-         -build.bin "tmp/bin/c4" \
+         -build.cmd "go build -tags=dev -o tmp/bin/games ." \
+         -build.bin "tmp/bin/games" \
          -build.exclude_dir "data,bin,tmp,deploy" \
          -build.include_ext "go,templ" \
          -misc.clean_on_exit "true"
@@ -75,7 +75,7 @@ tasks:
   run:
     desc: Build and run the server
     cmds:
-      - ./bin/c4
+      - ./bin/games
     deps:
       - build