fix(arcade): preserve game settings when returning to game selection

When users clicked "back to game selection", the clearRoomGameApi function
was sending both gameName: null AND gameConfig: null to the server. This
destroyed all saved game settings (like gameType, difficulty, etc.).

Now clearRoomGameApi only sends gameName: null and preserves gameConfig,
so settings persist when users select a game again.

Root cause discovered via comprehensive database-level logging that traced
the exact data flow through the system.

Fixes settings persistence bug in room mode.

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

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

apps/web/src/hooks/useRoomData.ts

@@ -634,6 +634,8 @@ export function useSetRoomGame() {
 
 /**
  * Clear/reset game for a room (host only)
+ * This only clears gameName (returns to game selection) but preserves gameConfig
+ * so settings persist when the user selects a game again.
  */
 async function clearRoomGameApi(roomId: string): Promise<void> {
   const response = await fetch(`/api/arcade/rooms/${roomId}/settings`, {
@@ -641,7 +643,7 @@ async function clearRoomGameApi(roomId: string): Promise<void> {
     headers: { 'Content-Type': 'application/json' },
     body: JSON.stringify({
       gameName: null,
-      gameConfig: null,
+      // DO NOT send gameConfig: null - we want to preserve settings!
     }),
   })