feat: extract and display membership gift counts in chat messages

client/app/dashboard/page.tsx

@@ -54,8 +54,8 @@ export default function DashboardPage() {
   }, [pollError, overlayStatus]);
 
   const superChats = useMemo(() => messages.filter(m => m.superChat), [messages]);
-  const newMembers = useMemo(() => messages.filter(m => m.membershipGiftPurchase), [messages]);
-  const regularMessages = useMemo(() => messages.filter(m => !m.superChat && !m.membershipGift), [messages]);
+  const newMembers = useMemo(() => messages.filter(m => m.membershipGift || m.membershipGiftPurchase), [messages]);
+  const regularMessages = useMemo(() => messages.filter(m => !m.superChat && !m.membershipGift && !m.membershipGiftPurchase), [messages]);
 
   return (
     <main className="dashboard">
@@ -462,7 +462,9 @@ function MemberItem({ message, isSelected, onSelect }: ChatItemProps) {
         <div className="memberItem__info">
           <span className="memberItem__author">{message.author}</span>
           <span className="memberItem__level">
-            {message.membershipLevel || 'New member'}
+            {message.membershipGiftPurchase && message.giftCount 
+              ? `Gifted ${message.giftCount} membership${message.giftCount > 1 ? 's' : ''}`
+              : message.membershipLevel || 'New member'}
           </span>
           <time className="memberItem__time">{new Date(message.publishedAt).toLocaleTimeString()}</time>
         </div>

client/app/overlay/page.tsx

@@ -89,12 +89,16 @@ export default function OverlayPage() {
               💰 {message.superChat.currency}{message.superChat.amount}
             </div>
           )}
-          {message.membershipGift && (
+          {(message.membershipGift || message.membershipGiftPurchase) && (
             <div className="overlay__membership">
-              🎁 New Member!
+              {message.membershipGiftPurchase && message.giftCount
+                ? `🎁 Gifted ${message.giftCount} Membership${message.giftCount > 1 ? 's' : ''}!`
+                : message.membershipGiftPurchase 
+                  ? '🎁 Gift Purchase' 
+                  : '🎁 New Member!'}
             </div>
           )}
-          {(!message.superChat && !message.membershipGift) && message.text && (
+          {(!message.superChat && !message.membershipGift && !message.membershipGiftPurchase) && message.text && (
             <p className="overlay__text">{message.text}</p>
           )}
         </div>