Refactor AppHeader and AreaTabs components for improved layout and functionality

- Adjusted AppHeader.vue for better spacing and alignment of elements, enhancing the overall visual appeal.
- Updated AreaTabs.vue to include a new 'Start' tab with a home icon, improving navigation options.
- Modified route handling in AreaTabs.vue to dynamically determine the current area based on the route.
- Changed default redirect destination in useAuth.ts to the root path for a more intuitive user experience.
- Enhanced index.vue to provide a more welcoming message and improved content presentation.

These changes aim to enhance user navigation and overall experience within the application.

app/components/navigation/AppHeader.vue

@@ -14,11 +14,8 @@ const { loggedIn } = useAuth()
       <!-- Left: Logo + Role Switcher -->
       <div class="flex items-center gap-6 md:gap-8">
         <NuxtLink to="/" class="flex-shrink-0 transition-transform hover:scale-105">
-          <img
-            src="/img/experimenta-logo-white.svg"
-            alt="experimenta Logo"
-            class="h-14 w-auto md:h-16 drop-shadow-lg"
-          />
+          <img src="/img/experimenta-logo-white.svg" alt="experimenta Logo"
+            class="h-14 w-auto md:h-16 drop-shadow-lg" />
         </NuxtLink>
 
         <!-- Role Switcher (Desktop only) -->
@@ -56,21 +53,17 @@ const { loggedIn } = useAuth()
 <style scoped>
 /* Custom experimenta styling */
 header {
-  background: linear-gradient(
-    135deg,
-    rgba(46, 16, 101, 0.98) 0%,
-    rgba(46, 16, 101, 0.95) 100%
-  );
+  background: linear-gradient(135deg,
+      rgba(46, 16, 101, 0.98) 0%,
+      rgba(46, 16, 101, 0.95) 100%);
   color: white;
 }
 
 /* Area tabs section with subtle gradient separator (no borders) */
 .area-tabs-section {
-  background: linear-gradient(
-    180deg,
-    rgba(255, 255, 255, 0.08) 0%,
-    rgba(255, 255, 255, 0.03) 100%
-  );
+  background: linear-gradient(180deg,
+      rgba(255, 255, 255, 0.08) 0%,
+      rgba(255, 255, 255, 0.03) 100%);
 }
 
 /* Mobile role section */