refactor(badges): fix text search and standardize field mapping

- Renamed 'default_qry_string' to 'default_qry_str' across the Badge interface, Dexie schema, and API logic to resolve 400 'Unauthorized search field' errors.
- Synchronized local Fast Path filtering with the correct database field names.
- Hardened the reactive search pattern in '+page.svelte' to ensure end-to-sync consistency between local and background results.

src/lib/ae_events/db_events.ts

@@ -159,7 +159,7 @@ export interface Badge {
 
     // data_json?: null|string;
 
-    default_qry_string?: null | string;
+    default_qry_str?: null | string;
 
     alert?: null | boolean;