refactor: improve router type safety and usage

- Add explicit Router type imports across views
- Replace $router type casting with proper typing
- Use $router.back() instead of $router.go(-1) for consistency
- Add proper route and router typings to components
- Clean up router navigation methods
- Fix router push/back method calls

This commit improves type safety and consistency in router usage across
the application's view components.

src/views/SearchAreaView.vue

@@ -131,6 +131,7 @@ const DEFAULT_ZOOM = 2;
 })
 export default class SearchAreaView extends Vue {
   $notify!: (notification: NotificationIface, timeout?: number) => void;
+  $router!: Router;
 
   isChoosingSearchBox = false;
   isNewMarkerSet = false;
@@ -219,7 +220,7 @@ export default class SearchAreaView extends Vue {
           },
           7000,
         );
-        (this.$router as Router).back();
+        this.$router.back();
       } catch (err) {
         this.$notify(
           {