feat: migrate authentication to Better Auth and update profile management logic

apps/frontend/src/lib/auth.tsx

@@ -20,6 +20,7 @@ type UpdateProfileParams = {
 };
 
 type UpdatePasswordParams = {
+  currentPassword?: string;
   password: string;
 };
 
@@ -172,11 +173,11 @@ export function AuthProvider({ children }: PropsWithChildren) {
         }
 
         return {
-          requiresEmailConfirmation: !signUpData?.session,
+          requiresEmailConfirmation: !signUpData || !('session' in signUpData),
         };
       },
       updateProfile: async ({ fullName }) => {
-        const { error } = await authClient.user.update({
+        const { error } = await authClient.updateUser({
           name: fullName,
         });
 
@@ -184,10 +185,11 @@ export function AuthProvider({ children }: PropsWithChildren) {
           throw error;
         }
       },
-      updatePassword: async ({ password }) => {
+      updatePassword: async ({ currentPassword, password }) => {
         // Better Auth uses changePassword for authenticated users
         const { error } = await authClient.changePassword({
           newPassword: password,
+          currentPassword: currentPassword || '',
           revokeOtherSessions: true,
         });