feat(mgmt/web): show the running game, and who decides when it ends

The host knows what it launched and, after a disconnect, that it is counting
down to closing it. None of that was visible: the console showed a stream with
no game, and a game on its way to being closed was something you found out about
afterwards.

The Dashboard gains a running-game card above the session card — box art matched
against the catalog it already fetches, so no new endpoint and no request on the
2 s status poll. "End now" means the two different things the row's state
implies: a live game ends by stopping its session (what then happens to the game
follows the policy — stopping a session is not licence to close a game), while
one already waiting out its reconnect window has no session left to stop and is
ended directly.

The settings card sits next to the display keep-alive policy because that is the
same question one step out: keep-alive decides how long a *display* outlives a
disconnect, this decides whether the *game* does. The copy says plainly what
`always` costs, that a drop is not someone pressing Stop, and that a display kept
forever is unaffected either way — the precedence rule that would otherwise
surprise someone. Where a build enforces nothing (macOS, no launch path) the
controls are shown disabled rather than hidden: "does nothing here" is
information.

Also fixes the story fixtures, which had gone stale against the `games[]` field
Phase 1 added, and adds a story for the state the card exists for — a game whose
client walked away.

web: build + tsc clean, biome-formatted; en/de messages complete.

web/messages/en.json

@@ -376,5 +376,28 @@
 	"store_phase_rolling_back": "Rolling back",
 	"store_phase_recording": "Recording provenance",
 	"store_phase_restarting": "Restarting the plugin runner",
-	"store_phase_done": "Done"
+	"store_phase_done": "Done",
+	"games_title": "Running games",
+	"games_state_launching": "Starting",
+	"games_state_running": "Running",
+	"games_state_exited": "Ended",
+	"games_state_grace": "Waiting for client",
+	"games_closing_in": "Its client is gone — closing in {time} unless it comes back",
+	"games_end_now": "End now",
+	"session_game_title": "When a game or a session ends",
+	"session_game_help": "A streaming session and the game it launched can share a fate. These settings are about the game; the keep-alive above is about the display, and the two have separate timers.",
+	"session_game_on_exit": "When the game exits",
+	"session_game_on_exit_help": "Quitting the game hands the client back to its own library instead of leaving it on your desktop. Turn this off if you stream the desktop and treat the game as incidental.",
+	"session_game_on_exit_end": "End the session",
+	"session_game_on_exit_keep": "Keep streaming",
+	"session_game_end_game": "When the session ends",
+	"session_game_end_game_help": "Whether stopping (or losing) a session also closes the game it launched. Never applies to a game you started yourself — only one this host launched for the session.",
+	"session_game_end_keep": "Leave it running",
+	"session_game_end_on_quit": "Close it on Stop",
+	"session_game_end_always": "Always close it",
+	"session_game_always_warning": "Closing a game costs whatever it had not saved. The host asks it to close first and only forces the issue if it refuses — but a network drop is not someone pressing Stop, so a dropped client gets the reconnect window below before anything happens. A display kept forever stays up regardless; this setting governs the game, not the screen.",
+	"session_game_grace": "Reconnect window",
+	"session_game_grace_help": "How long a client that vanished has to come back before its game is closed. The console shows the countdown, and reconnecting cancels it.",
+	"session_game_saved": "Session and game settings saved",
+	"session_game_inert": "This host has no way to launch games, so these settings do nothing here"
 }