[Visualize] Fix inspector download filename issue when saving in-place (

#72605)

Co-authored-by: Elastic Machine <elasticmachine@users.noreply.github.com>

src/plugins/visualizations/public/embeddable/visualize_embeddable.ts

@@ -157,7 +157,7 @@ export class VisualizeEmbeddable extends Embeddable<VisualizeInput, VisualizeOut
     if (!adapters) return;
 
     return this.deps.start().plugins.inspector.open(adapters, {
-      title: this.getTitle() || '',
+      title: this.getTitle() || this.title || '',
     });
   };
 
@@ -215,8 +215,17 @@ export class VisualizeEmbeddable extends Embeddable<VisualizeInput, VisualizeOut
       dirty = true;
     }
 
+    // propagate the title to the output embeddable
+    // but only when the visualization is in edit/Visualize mode
+    if (!this.parent && this.vis.title !== this.output.title) {
+      this.updateOutput({ title: this.vis.title });
+    }
+
+    // Keep title depending on the output Embeddable to decouple the
+    // visual appearance of the title and the actual title content (useful in Dashboard)
     if (this.output.title !== this.title) {
       this.title = this.output.title;
+
       if (this.domNode) {
         this.domNode.setAttribute('data-title', this.title || '');
       }