fix: use same-origin default for auth URL instead of localhost

Avoids ERR_CONNECTION_REFUSED in deployed environments where
VITE_AUTH_URL is not set at build time. Empty-string fallback
routes auth requests to same origin, which the HTTPRoute forwards
to the auth service.

cc @cpfarhood

Co-Authored-By: Paperclip <noreply@paperclip.ing>

src/lib/auth-client.ts

@@ -1,8 +1,36 @@
 import { createAuthClient } from "better-auth/react"
+import type { BetterFetchPlugin } from "@better-fetch/fetch"
+
+/**
+ * Maps 'name' -> 'display_name' in register requests to match the API's RegisterRequest schema.
+ */
+const displayNameMapper: BetterFetchPlugin = {
+  id: "display-name-mapper",
+  name: "display-name-mapper",
+  hooks: {
+    onRequest: async (context) => {
+      const url = typeof context.url === "string" ? context.url : context.url.pathname
+      if (
+        url.endsWith("/auth/register") &&
+        context.method === "POST" &&
+        context.body &&
+        "name" in context.body
+      ) {
+        context.body = {
+          ...context.body,
+          display_name: context.body.name as string,
+          name: undefined,
+        }
+      }
+      return context
+    },
+  },
+}
 
 export const authClient = createAuthClient({
-  baseURL: import.meta.env.VITE_AUTH_URL ?? "http://localhost:3001",
+  baseURL: import.meta.env.VITE_AUTH_URL || "",
   basePath: "/auth",
+  fetchPlugins: [displayNameMapper],
 })
 
 export const { useSession, signIn, signUp, signOut } = authClient