Add preliminary support for multiple card games

A game-type registry (server/src/tavolo/games.py) is now the single
source of truth for the games the platform can host; only scopone
scientifico is registered so far. GET /api/game-types exposes it for the
lobby's new game dropdown, and POST /api/games accepts a validated
game_type (default scopone_scientifico) which is carried on the live
GameState and onto each finished Match row (new indexed column,
migration 1_20260916235833_update), so statistics can be scoped per
game: /api/me/matches and /api/leaderboard take an optional game_type
filter and every serialized match includes its game_type.

Game states serialized before this change still load with the default
game type.

server/tests/test_store.py

@@ -25,6 +25,24 @@ class InMemoryGameStoreTest(unittest.TestCase):
         self.assertEqual(16, loaded.target_score)
         self.assertEqual(["alice", "bob"], [p.sub for p in loaded.players])
 
+    @async_test
+    async def test_game_type_roundtrip_and_default(self) -> None:
+        store = InMemoryGameStore()
+        state = engine.create_game(
+            "g1b", "CODE1B", "alice", "alice", game_type="scopone_scientifico"
+        )
+        await store.save(state)
+        loaded = await store.load("g1b")
+        assert loaded is not None
+        self.assertEqual("scopone_scientifico", loaded.game_type)
+
+        # States serialized before game types existed load with the default.
+        legacy = state.to_json()
+        del legacy["game_type"]
+        from tavolo.game.state import GameState
+
+        self.assertEqual("scopone_scientifico", GameState.from_json(legacy).game_type)
+
     @async_test
     async def test_load_missing_returns_none(self) -> None:
         store = InMemoryGameStore()