feat(host): game-mode integration + dedicated game sessions

Implements design/gamemode-and-dedicated-sessions.md (Parts A1-A5 + B0-B2):
reconciles the merged display-management registry with session-mobile
Bazzite/SteamOS hosts and adds a per-launch dedicated gamescope mode.

- A1 DisplayOwnership {Owned,External,SessionManaged} + poolable_now(): the
  registry pools only what it owns, so gamescope managed/attach outputs are no
  longer double-owned by the registry AND the gamescope restore worker (fixes
  the game-mode-reconnect stale-node wedge).
- A2 validated reuse: (backend,mode,launch,epoch) reuse key + kept_display_alive
  liveness probe + reused_gen/mark_failed on a reused-display first-frame failure.
- A3 policy-driven managed restore (keep_alive replaces the hardcoded 5s debounce;
  forever = held = gaming-rig truthful) + crash-restore persist + SIGKILL teardown
  (kill_unit, applied to our transient unit AND the autologin stop -- validated
  live on .181 to avoid the F44 GPU-context leak).
- A4 session epoch: observe_session_instance bumps the epoch + invalidate_backend
  on a desktop-compositor instance change; gamescope spawns are exempt.
- A5 per-spawn log + PID-scoped gamescope node discovery.
- B0 game_session {auto,dedicated} policy (top-level, preset-orthogonal) +
  pick_gamescope_mode dedicated_launch + steam -silent command shaping.
- B1 free the autologin Steam before a dedicated Steam spawn (single-instance).
- B2 game-exit -> APP_EXITED_CLOSE_CODE (0x52) clean session end.

Adversarially reviewed (11 findings fixed). Validated on glass (.181 Bazzite F44,
RTX 4090): dedicated spawn streams a real game smoothly; keep-alive reuse; the
SIGKILL fix avoids the F44 vkCreateDevice leak. Workspace green
(build / test --workspace / clippy -D warnings / fmt), OpenAPI + C header
regenerated, web console tsc + vite build green. clients/probe: bump the
no-video timeout 8s->45s for gamescope cold starts.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

api/openapi.json

@@ -10,7 +10,7 @@
       "name": "MIT OR Apache-2.0",
       "identifier": "MIT OR Apache-2.0"
     },
-    "version": "0.7.4"
+    "version": "0.8.0"
   },
   "paths": {
     "/api/v1/clients": {
@@ -2240,6 +2240,10 @@
         "type": "object",
         "description": "The user-facing display-management policy — what `display-settings.json` holds and what the mgmt\nAPI GETs/PUTs. When [`preset`](Self::preset) is not [`Preset::Custom`] the explicit fields are\nignored (the console writes one or the other); [`effective`](Self::effective) resolves both to a\nsingle [`EffectivePolicy`].",
         "properties": {
+          "game_session": {
+            "$ref": "#/components/schemas/GameSession",
+            "description": "How a game-launching session is served (`design/gamemode-and-dedicated-sessions.md` §5.2).\nOrthogonal to `preset`/lifecycle — preserved across preset changes; `#[serde(default)]` = `Auto`\nso existing `display-settings.json` files are untouched."
+          },
           "identity": {
             "$ref": "#/components/schemas/Identity"
           },
@@ -2399,6 +2403,14 @@
           }
         }
       },
+      "GameSession": {
+        "type": "string",
+        "description": "How a session that **launches a game** (a library id on the Hello / apps.json / Decky pin) is\nserved (`design/gamemode-and-dedicated-sessions.md` §5.2). Orthogonal to the preset/lifecycle axes\n— a top-level [`DisplayPolicy`] field, NOT part of [`EffectivePolicy`], so a preset never clobbers\nit. Linux-only in effect (a launching Windows session opens into the one desktop).",
+        "enum": [
+          "auto",
+          "dedicated"
+        ]
+      },
       "GpuState": {
         "type": "object",
         "description": "Full GPU-selection state for the console: inventory, the persisted preference, what the next\nsession will use, and what is in use right now.",