fix: sync DOM widget values to widgetValueStore on registration (#9166)

## Summary

Override `setNodeId` in `BaseDOMWidgetImpl` to sync the DOM-resolved
value into the widget value store, fixing empty system prompts in Vue
nodes (Nodes 2.0).

## Changes

- **What**: DOM widgets (e.g. textarea for Gemini system_prompt) resolve
their value through `options.getValue()` / DOM elements, not
`_state.value`. When `BaseWidget.setNodeId` registers with the store, it
spreads `_state.value` which is `undefined` for DOM widgets. The
override captures the DOM-resolved value before registration and syncs
it into the store afterward — keeping the fix in the DOM widget layer
where the mismatch originates, leaving `BaseWidget` unchanged.

## Review Focus

- Whether capturing `this.value` before `super.setNodeId()` and writing
it after is the right sequencing
- Whether this correctly handles all DOM widget subtypes
(`DOMWidgetImpl`, `ComponentWidgetImpl`)

Supersedes #9164

┆Issue is synchronized with this [Notion
page](https://www.notion.so/PR-9166-fix-sync-DOM-widget-values-to-widgetValueStore-on-registration-3116d73d3650816f8cece866a9272baa)
by [Unito](https://www.unito.io)

src/scripts/domWidgetStore.test.ts

@@ -0,0 +1,48 @@
+import { createTestingPinia } from '@pinia/testing'
+import { setActivePinia } from 'pinia'
+import { beforeEach, describe, expect, it } from 'vitest'
+
+import { LGraph, LGraphNode } from '@/lib/litegraph/src/litegraph'
+import { DOMWidgetImpl } from '@/scripts/domWidget'
+import { useWidgetValueStore } from '@/stores/widgetValueStore'
+
+describe('DOMWidgetImpl store integration', () => {
+  let graph: LGraph
+  let node: LGraphNode
+  let store: ReturnType<typeof useWidgetValueStore>
+
+  beforeEach(() => {
+    setActivePinia(createTestingPinia({ stubActions: false }))
+    store = useWidgetValueStore()
+    graph = new LGraph()
+    node = new LGraphNode('TestNode')
+    node.id = 1
+    graph.add(node)
+  })
+
+  it('registers DOM-resolved value in store via setNodeId', () => {
+    const defaultValue = 'You are an expert image-generation engine.'
+    const element = document.createElement('textarea')
+    element.value = defaultValue
+
+    const widget = new DOMWidgetImpl({
+      node,
+      name: 'system_prompt',
+      type: 'customtext',
+      element,
+      options: {
+        getValue: () => element.value as string,
+        setValue: (v: string) => {
+          element.value = v
+          const state = store.getWidget(graph.id, node.id, 'system_prompt')
+          if (state) state.value = v
+        }
+      }
+    })
+
+    widget.setNodeId(node.id)
+
+    const state = store.getWidget(graph.id, node.id, 'system_prompt')
+    expect(state?.value).toBe(defaultValue)
+  })
+})