chore(logging): normalize wakeup flow observability with timings and summaries

Standardize console prefixes across scheduler, push, refresh, register,
auth, and debug endpoints. Add pass-level scheduler summaries, elapsed-time
logs, and masked-token-only push failure messages while reducing per-device
noise in scheduler loops.

src/util/formatElapsed.ts

@@ -0,0 +1,17 @@
+/** Human-readable duration for console logs (e.g. 842ms, 2.1s). */
+export function formatElapsedMs(elapsedMs: number): string {
+  if (elapsedMs < 1000) {
+    return `${Math.round(elapsedMs)}ms`;
+  }
+  return `${(elapsedMs / 1000).toFixed(1)}s`;
+}
+
+export function errorMessage(err: unknown): string {
+  if (err instanceof Error && err.message.length > 0) {
+    return err.message;
+  }
+  if (typeof err === "string" && err.length > 0) {
+    return err;
+  }
+  return "Unknown error";
+}