feat: add friend invite modal to all game modes

- New InviteFriendsModal component: fetches friends list, sends game
  invite via notification socket with per-friend sent confirmation
- Multiplayer (GamePage): invite button below copy link, hidden after game ends
- Correspondence: invite button in waiting-for-opponent card
- Tournament: invite button for joined players while status is waiting
- Invite links use mode-prefixed codes (correspondence/XYZ, tournament/XYZ)
  so notification "Beitreten" button navigates to the correct page

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

client/src/components/game/GamePage.tsx

@@ -23,6 +23,7 @@ import type { ClearPremoves } from "react-chessboard";
 import { Chessboard } from "react-chessboard";
 
 import { API_URL, APP_URL } from "@/config";
+import InviteFriendsModal from "@/components/InviteFriendsModal";
 import { io } from "socket.io-client";
 
 import { lobbyReducer, squareReducer } from "./reducers";
@@ -654,6 +655,11 @@ export default function GamePage({ initialLobby }: { initialLobby: Game }) {
                 </div>
               </div>
             </div>
+            {!lobby.endReason && (
+              <div className="mt-1 flex justify-end">
+                <InviteFriendsModal gameCode={initialLobby.code!} label="Freunde einladen" />
+              </div>
+            )}
             <div className="h-32 w-full overflow-y-scroll" ref={moveListRef}>
               <table className="table-compact table w-full">
                 <tbody>{getMoveListHtml()}</tbody>