docs(website): tabbed per-OS install instructions with auto-detection

Split the install page into Linux / macOS / Windows tabs so each visitor
sees only the commands that apply to them. A small InstallOsDetect component
seeds Starlight's synced-tabs localStorage key with the detected OS before
the tabs render, so the matching tab is pre-selected with no flash — and
only when the visitor hasn't already picked a tab, so a manual choice wins.

- installation.md -> .mdx (needs the Tabs/TabItem components)
- detection covers navigator.userAgentData and the legacy platform/UA
  fallbacks; unknown platforms fall through to the default tab
- piggybacks on Starlight's own restore mechanism rather than re-implementing
  tab switching

website/src/components/InstallOsDetect.astro

@@ -0,0 +1,39 @@
+---
+// Pre-selects the install-instructions tab that matches the visitor's OS.
+//
+// This piggybacks on Starlight's synced-tabs mechanism rather than fighting it.
+// A `<Tabs syncKey="install-os">` restores its active tab from
+// `localStorage["starlight-synced-tabs__install-os"]` via a script Starlight
+// inlines *before* the tabs render (so there is no flash of the wrong tab). All
+// we do here is seed that same key with the detected OS — and only when the
+// visitor has not already chosen a tab themselves, so a manual choice always wins.
+//
+// Requirements (mirrors Starlight's own restore script):
+//   - Must be placed *before* the `<Tabs>` it targets, so this runs first.
+//   - The stored value must equal one of the TabItem labels exactly
+//     ("Linux" / "macOS" / "Windows").
+//   - Inlined, so it runs during parse before the restore script's
+//     connectedCallback and paint.
+---
+
+<script is:inline>
+	(() => {
+		const KEY = 'starlight-synced-tabs__install-os';
+		try {
+			// A previously stored value is a deliberate user choice — never override it.
+			if (localStorage.getItem(KEY)) return;
+			const data = navigator.userAgentData;
+			const platform = ((data && data.platform) || navigator.platform || '').toLowerCase();
+			const ua = (navigator.userAgent || '').toLowerCase();
+			let os = null;
+			if (platform.includes('win') || ua.includes('windows')) os = 'Windows';
+			else if (platform.includes('mac') || ua.includes('mac os x')) os = 'macOS';
+			else if (platform.includes('linux') || platform.includes('x11') || ua.includes('linux'))
+				os = 'Linux';
+			if (os) localStorage.setItem(KEY, os);
+		} catch (e) {
+			// localStorage unavailable (private mode, blocked cookies, …) — silently
+			// fall back to the default first tab. No detection, no harm.
+		}
+	})();
+</script>