feat(home): label the top-bar online counter with the hotel name

Use the shared LiveOnlineCount so the landing top bar reads "0 <hotel> online"
instead of a bare number, and drop the hotel name from the left (the logo
stays) so it is not shown twice.

src/components/auth/auth-top-bar.tsx

@@ -1,28 +1,27 @@
 import Image from "next/image";
 import { Clock } from "@/components/clock";
 import { LanguageSwitcher } from "@/components/language-switcher";
-import { LiveOnlineCounter } from "@/components/live-online-counter";
+import { LiveOnlineCount } from "@/components/live-online-count";
 import { Reveal } from "@/components/motion-reveal";
 import { ThemeSwitcher } from "@/components/theme-switcher";
 import { ICON_LIGHTHOUSE } from "@/lib/site-icons";
 
 /**
  * Shared frosted-glass top bar for the public pages (index, login, register).
- * Shows the hotel logo + name on the left and the live online count, language /
+ * Shows the hotel logo on the left and the live "online" counter (which carries
- * clock / theme controls on the right. One component so every public page keeps
+ * the hotel name), language / clock / theme controls on the right. One
- * the exact same chrome instead of duplicating it.
+ * component so every public page keeps the exact same chrome instead of
+ * duplicating it.
  */
 export function AuthTopBar({
 	hotelName,
 	logo,
 	online,
-	showHotelName = true,
 }: {
 	hotelName: string;
 	logo?: string | null;
 	/** Live online count; omit (or pass null) to hide the indicator. */
 	online?: number | null;
-	showHotelName?: boolean;
 }) {
 	return (
 		<Reveal>
@@ -71,14 +70,6 @@ export function AuthTopBar({
 							/>
 						</div>
 					)}
-					{showHotelName ? (
-						<span
-							className="hidden truncate text-sm font-extrabold uppercase tracking-wide sm:inline"
-							style={{ color: "var(--color-text-readable)" }}
-						>
-							{hotelName}
-						</span>
-					) : null}
 				</div>
 				<div
 					className="flex items-center gap-1 sm:gap-1.5 rounded-full px-1.5 py-1 sm:px-2"
@@ -108,7 +99,7 @@ export function AuthTopBar({
 									color: "var(--color-primary-readable, var(--color-primary))",
 								}}
 							>
-								<LiveOnlineCounter initial={online} />
+								<LiveOnlineCount initial={online} hotelName={hotelName} />
 							</span>
 						</span>
 					)}