fix(arcade): read nested gameConfig correctly when creating sessions

The session initialization was looking for settings at the wrong level:
- Was reading: room.gameConfig.gameType (undefined, falls back to default)
- Should read: room.gameConfig.matching.gameType (saved value)

gameConfig is structured as:
{
  "matching": { "gameType": "...", "difficulty": ..., "turnTimer": ... },
  "memory-quiz": { "selectedCount": ..., "displayTime": ..., ... }
}

This caused the session to be created with default settings even though
the settings were saved in the database. The client would load the correct
settings from roomData.gameConfig, but then the socket would immediately
overwrite them with the session's default state.

Now properly accesses the nested config for each game type.

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

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

apps/web/src/socket-server.ts

@@ -84,16 +84,20 @@ export function initializeSocketServer(httpServer: HTTPServer) {
               // Different games have different initial configs
               let initialState: any
               if (room.gameName === 'matching') {
+                // Access nested gameConfig: { matching: { gameType, difficulty, turnTimer } }
+                const matchingConfig = (room.gameConfig as any)?.matching || {}
                 initialState = validator.getInitialState({
-                  difficulty: (room.gameConfig as any)?.difficulty || 6,
-                  gameType: (room.gameConfig as any)?.gameType || 'abacus-numeral',
-                  turnTimer: (room.gameConfig as any)?.turnTimer || 30,
+                  difficulty: matchingConfig.difficulty || 6,
+                  gameType: matchingConfig.gameType || 'abacus-numeral',
+                  turnTimer: matchingConfig.turnTimer || 30,
                 })
               } else if (room.gameName === 'memory-quiz') {
+                // Access nested gameConfig: { 'memory-quiz': { selectedCount, displayTime, selectedDifficulty } }
+                const memoryQuizConfig = (room.gameConfig as any)?.['memory-quiz'] || {}
                 initialState = validator.getInitialState({
-                  selectedCount: (room.gameConfig as any)?.selectedCount || 5,
-                  displayTime: (room.gameConfig as any)?.displayTime || 2.0,
-                  selectedDifficulty: (room.gameConfig as any)?.selectedDifficulty || 'easy',
+                  selectedCount: memoryQuizConfig.selectedCount || 5,
+                  displayTime: memoryQuizConfig.displayTime || 2.0,
+                  selectedDifficulty: memoryQuizConfig.selectedDifficulty || 'easy',
                 })
               } else {
                 // Fallback for other games