Start game via WebSocket intent (#3794)

## Description:

Replaces the HTTP POST /api/start_game/:id endpoint with a WebSocket
intent, making private game start consistent with how kick_player and
update_game_config already work. Also verifies that only the lobby
creator can start a game.

## Please complete the following:

- [x] I have added screenshots for all UI updates
- [x] I process any text displayed to the user through translateText()
and I've added it to the en.json file
- [x] I have added relevant tests to the test directory
- [x] I confirm I have thoroughly tested these changes and take full
responsibility for any bugs introduced

## Please put your Discord username so you can be contacted if a bug or
regression is found:

evan

src/core/Schemas.ts

@@ -50,7 +50,8 @@ export type Intent =
   | DeleteUnitIntent
   | KickPlayerIntent
   | TogglePauseIntent
-  | UpdateGameConfigIntent;
+  | UpdateGameConfigIntent
+  | StartGameIntent;
 
 export type AttackIntent = z.infer<typeof AttackIntentSchema>;
 export type CancelAttackIntent = z.infer<typeof CancelAttackIntentSchema>;
@@ -84,6 +85,7 @@ export type TogglePauseIntent = z.infer<typeof TogglePauseIntentSchema>;
 export type UpdateGameConfigIntent = z.infer<
   typeof UpdateGameConfigIntentSchema
 >;
+export type StartGameIntent = z.infer<typeof StartGameIntentSchema>;
 
 export type Turn = z.infer<typeof TurnSchema>;
 export type GameConfig = z.infer<typeof GameConfigSchema>;
@@ -453,6 +455,10 @@ export const UpdateGameConfigIntentSchema = z.object({
   config: GameConfigSchema.partial(),
 });
 
+export const StartGameIntentSchema = z.object({
+  type: z.literal("start_game"),
+});
+
 const IntentSchema = z.discriminatedUnion("type", [
   AttackIntentSchema,
   CancelAttackIntentSchema,
@@ -478,6 +484,7 @@ const IntentSchema = z.discriminatedUnion("type", [
   KickPlayerIntentSchema,
   TogglePauseIntentSchema,
   UpdateGameConfigIntentSchema,
+  StartGameIntentSchema,
 ]);
 
 // StampedIntent = Intent with server-stamped clientID (used in turns and execution)