Feat: Add Badges for Vue Nodes (#6243)

## Summary

Adds the badges to the header for Vue nodes.


## Review Focus

Design, mostly. Any structures here I'm not handling but should be?

## Screenshots
<img width="1514" height="642" alt="image"
src="https://github.com/user-attachments/assets/387fd2f6-bb4b-4fee-b273-6166a52a3552"
/>

<!-- Add screenshots or video recording to help explain your changes -->

┆Issue is synchronized with this [Notion
page](https://www.notion.so/PR-6243-Feat-Add-Badges-for-Vue-Nodes-2956d73d36508184a250d67b127ed4b1)
by [Unito](https://www.unito.io)

---------

Co-authored-by: github-actions <github-actions@github.com>

src/composables/graph/useGraphNodeManager.ts

@@ -20,6 +20,7 @@ import type { WidgetValue } from '@/types/simplifiedWidget'
 
 import type {
   LGraph,
+  LGraphBadge,
   LGraphNode,
   LGraphTriggerAction,
   LGraphTriggerEvent,
@@ -51,6 +52,8 @@ export interface VueNodeData {
   mode: number
   selected: boolean
   executing: boolean
+  apiNode?: boolean
+  badges?: (LGraphBadge | (() => LGraphBadge))[]
   subgraphId?: string | null
   widgets?: SafeWidgetData[]
   inputs?: INodeInputSlot[]
@@ -117,7 +120,7 @@ export function useGraphNodeManager(graph: LGraph): GraphNodeManager {
   }
 
   // Extract safe data from LiteGraph node for Vue consumption
-  const extractVueNodeData = (node: LGraphNode): VueNodeData => {
+  function extractVueNodeData(node: LGraphNode): VueNodeData {
     // Determine subgraph ID - null for root graph, string for subgraphs
     const subgraphId =
       node.graph && 'id' in node.graph && node.graph !== node.graph.rootGraph
@@ -192,6 +195,9 @@ export function useGraphNodeManager(graph: LGraph): GraphNodeManager {
       node.constructor?.name ||
       'Unknown'
 
+    const apiNode = node.constructor?.nodeData?.api_node ?? false
+    const badges = node.badges
+
     return {
       id: String(node.id),
       title: typeof node.title === 'string' ? node.title : '',
@@ -200,6 +206,8 @@ export function useGraphNodeManager(graph: LGraph): GraphNodeManager {
       selected: node.selected || false,
       executing: false, // Will be updated separately based on execution state
       subgraphId,
+      apiNode,
+      badges,
       hasErrors: !!node.has_errors,
       widgets: safeWidgets,
       inputs: node.inputs ? [...node.inputs] : undefined,