fix: Make Register links properly navigate to registration form

- Initialize isRegistering state from URL query parameter (mode=register)
- Update URL when toggling between login/register modes for shareable links
- Remove duplicate useRoute() calls in login/register handlers
- Ensures all Register buttons throughout the app show the registration form

This fixes the issue where Register links navigated to /login?mode=register
but the login page ignored the mode parameter and showed the login form instead.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

pages/login.vue

@@ -169,6 +169,9 @@
 </template>
 
 <script setup lang="ts">
+const route = useRoute()
+const router = useRouter()
+
 const username = ref('')
 const password = ref('')
 const confirmPassword = ref('')
@@ -177,7 +180,9 @@ const firstName = ref('')
 const lastName = ref('')
 const error = ref('')
 const loading = ref(false)
-const isRegistering = ref(false)
+
+// Initialize from URL query parameter
+const isRegistering = ref(route.query.mode === 'register')
 
 const passwordRequirements = computed(() => ({
   minLength: password.value.length >= 8,
@@ -206,6 +211,15 @@ function toggleMode() {
   email.value = ''
   firstName.value = ''
   lastName.value = ''
+
+  // Update URL to reflect mode
+  const query = { ...route.query }
+  if (isRegistering.value) {
+    query.mode = 'register'
+  } else {
+    delete query.mode
+  }
+  router.replace({ query })
 }
 
 async function handleLogin() {
@@ -223,7 +237,6 @@ async function handleLogin() {
 
     if (response.success) {
       // Check if there's a redirect parameter
-      const route = useRoute()
       const redirect = route.query.redirect as string
       await navigateTo(redirect || '/')
     }
@@ -265,7 +278,6 @@ async function handleRegister() {
 
     if (response.success) {
       // Check if there's a redirect parameter
-      const route = useRoute()
       const redirect = route.query.redirect as string
       await navigateTo(redirect || '/')
     }