Extend badge color settings to cover integration types too

Settings → DNS Badges only let you customize DNS provider badge
colors. Expand it into a general Badges page with a second section
for the six integration types (Tailscale, Proxmox, Synology,
Semaphore, Gitea, Dockhand), applied to the type badge in the Manage
integrations table.

- New integrationColors key on AppSettings, stored/merged the same way
  as providerColors via the existing settingsStore.
- New GET /api/settings/integration-colors — non-secret, any signed-in
  user, mirroring /provider-colors — so the badge color can be read
  without needing admin access to the full settings payload.
- Renamed DnsBadgeSettings.tsx -> BadgeSettings.tsx (route
  /settings/dns-badges -> /settings/badges, sub-nav label "DNS
  Badges" -> "Badges") with both color sections saved together.

Verified end-to-end against the real dev server: PUT persists
integration colors independently of provider colors, the public
integration-colors endpoint reflects updates immediately, and the
change is audit-logged.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

web/src/App.tsx

@@ -13,7 +13,7 @@ import ServerDetail from "./pages/ServerDetail";
 import Integrations from "./pages/Integrations";
 import Settings from "./pages/Settings";
 import NotificationSettings from "./pages/settings/NotificationSettings";
-import DnsBadgeSettings from "./pages/settings/DnsBadgeSettings";
+import BadgeSettings from "./pages/settings/BadgeSettings";
 import CacheSettings from "./pages/settings/CacheSettings";
 import AppShell from "./layout/AppShell";
 
@@ -95,7 +95,7 @@ export default function App() {
         >
           <Route index element={<Navigate to="notifications" replace />} />
           <Route path="notifications" element={<NotificationSettings />} />
-          <Route path="dns-badges" element={<DnsBadgeSettings />} />
+          <Route path="badges" element={<BadgeSettings />} />
           <Route path="cache" element={<CacheSettings />} />
         </Route>
       </Routes>