Phase 2.4: multiplayer.php now routes to correct table for all games
(chess→matches, domino→domino_matches, backgammon→backgammon_matches,
ludo→ludo_matches). Validates game_key against allowed list.

Phase 4.3: handleServerState in backgammon only copies specific fields
(state, dice, turn, gameOver) instead of Object.assign which clobbered
local methods and properties.
Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>