feat(game): canonical gameId in POST /api/v1/admin/init

Engine no longer mints its own game UUID. The orchestrator (backend)
generates the game UUID at game-create time and passes it in the
admin/init request body as the required `gameId` field, so the value
that names the engine container and host bind-mount directory also
ends up inside the engine's state.json.

The engine rejects the zero UUID with 400 and any init that conflicts
with an existing state.json with 409 (a second init on the same gameId
is also a conflict; full idempotency is not part of the contract).

Updates rest.InitRequest, openapi.yaml (schema + 409 response),
controller.GenerateGame/NewGame/buildGameOnMap signatures, the engine
HTTP handler/executor, the backend runtime worker, and the relevant
unit and contract tests. Documentation in game/README.md,
docs/ARCHITECTURE.md, backend/README.md, and backend/docs/{runtime,flows}.md
is updated in the same patch.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

game/internal/controller/init_errors.go

@@ -0,0 +1,14 @@
+package controller
+
+import "errors"
+
+// ErrGameInitNilUUID is returned by GenerateGame when the supplied
+// game UUID is the zero value. The HTTP layer maps it to 400.
+var ErrGameInitNilUUID = errors.New("game init: gameId must not be the zero UUID")
+
+// ErrGameAlreadyInit is returned by GenerateGame when the engine
+// storage directory already contains a state.json. The HTTP layer
+// maps it to 409. Repeated init on the same gameId is intentionally
+// rejected rather than treated as a no-op; full idempotency is not
+// part of the contract.
+var ErrGameAlreadyInit = errors.New("game init: game already initialized")