feat(demo): list the demo's sign-ins on /beta and link into it per game

The public demo (demo.runicgateway.com) is live, so the site's demo slot
can be switched on. Two decisions of record, PLAN.md §15:

- D82: /beta reads the demo's GET /api/v1/public/settings demo.logins on
  the server, caches it for a minute, and lists the published sign-ins
  with the next wipe. If the demo does not answer, /beta links to its
  sign-in page instead. Gate two reads as in place.
- D83: a deep link into a game module's pages carries data-demo-game, and
  applyBrand.mjs fills it only for games in the new mounted demoGames
  list. The Rust features gain links into /rust and /rust/servers/demo.
  UO's stay hidden on a Rust demo.

The public-demo not-built entry is removed, because its resolvedBy was met.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01E14m6SuuY6i1vASFeGDBeY

src/data/capabilities.mjs

@@ -51,7 +51,7 @@
  * ---------------------------------------------------------------------------------------
  * `demoPath` — DEEP LINKS THAT ONLY EXIST WHEN A DEMO DOES
  * ---------------------------------------------------------------------------------------
- * §15/D12 keeps the public demo out of scope while requiring the site to gain one by way of
+ * §15/D12 kept the public demo out of scope while requiring the site to gain one by way of
  * a line in a mounted `brand.json`. `brand.json`'s own comment promised `/features/` a
  * "per-capability affordance" that had never been defined; D25 defines it as a deep link
  * per capability that has a stable public route, appended to the mounted `demoUrl` by
@@ -63,8 +63,10 @@
  * behind a Team id no static page can know. Paths are read from the real route tables —
  * core's `client/src/App.jsx` and `module-uo`'s `client/src/entry.jsx` on `main` — never
  * guessed. A module's public pages are namespaced under its own id, so a route into the
- * Ultima Online module is `/uo/…`: the demo §15 plans runs that game, which is why the
- * Teams item points there and the module-supplied items in `games/` carry their own.
+ * Ultima Online module is `/uo/…`. A link into a module's pages names that game
+ * (`demoGame` here, the `game` prop of `GameFeatures` for the lists in `games/`), and it
+ * shows only when the mounted `demoGames` lists it (D83): the demo that exists runs Rust,
+ * so the Teams item's `/uo/guilds` stays hidden there and Rust's clans link stands in.
  */
 
 import { games } from './games/index.mjs';
@@ -82,6 +84,7 @@ import { games } from './games/index.mjs';
  * @property {string[]} [caps]   Module capability slugs, on a game's own items (`games/`) only.
  * @property {boolean} [needsModule] Core machinery a module has to populate (D24).
  * @property {string} [demoPath] A stable public route, deep-linked into a demo (D25).
+ * @property {string} [demoGame] The game module whose pages `demoPath` is in, if any (D83).
  *
  * @typedef {object} CapabilityGroup
  * @property {string} id
@@ -116,6 +119,7 @@ const community = {
       label: 'Teams',
       needsModule: true,
       demoPath: '/uo/guilds',
+      demoGame: 'uo',
       detail:
         'A roster, a leader, a private forum, its own notification streams and a Discord ' +
         'voice channel, all hanging off one group. Core owns every part of that machinery ' +