fix: sign-out redirect and Google OAuth user creation

- signOut() cleared the cookie but never navigated, leaving the user
  stuck on the current page — redirect to /login on success.
- better-auth's core "image" field had no matching Drizzle column
  (schema uses avatarUrl), so every OAuth signup threw
  BetterAuthError: unable_to_create_user, silently failing before a
  session was ever created. Remap via user.fields.image.

apps/web/components/layout/nav.tsx

@@ -1,7 +1,7 @@
 "use client";
 
 import Link from "next/link";
-import { usePathname } from "next/navigation";
+import { usePathname, useRouter } from "next/navigation";
 import { BookOpen, Calendar, Package, ChefHat, User, Rss, FolderOpen, ShoppingCart, Shield, Search, Compass, Menu } from "lucide-react";
 import { cn } from "@/lib/utils";
 import { Button, buttonVariants } from "@/components/ui/button";
@@ -36,6 +36,7 @@ const NAV_ITEMS = [
 
 export function Nav() {
   const pathname = usePathname();
+  const router = useRouter();
   const { data: session } = authClient.useSession();
   const isAdmin = (session?.user as { role?: string } | undefined)?.role === "admin";
   const t = useTranslations("nav");
@@ -124,7 +125,18 @@ export function Nav() {
                 </>
               )}
               <DropdownMenuSeparator />
-              <DropdownMenuItem onClick={() => authClient.signOut()}>
+              <DropdownMenuItem
+                onClick={() => {
+                  void authClient.signOut({
+                    fetchOptions: {
+                      onSuccess: () => {
+                        router.push("/login");
+                        router.refresh();
+                      },
+                    },
+                  });
+                }}
+              >
                 {t("signOut")}
               </DropdownMenuItem>
             </DropdownMenuContent>

apps/web/lib/auth/server.ts

@@ -100,6 +100,9 @@ export const auth = betterAuth({
   },
 
   user: {
+    fields: {
+      image: "avatarUrl",
+    },
     additionalFields: {
       role: {
         type: "string",