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/domWidget.ts

@@ -7,6 +7,7 @@ import {
   LegacyWidget,
   LiteGraph
 } from '@/lib/litegraph/src/litegraph'
+import type { NodeId } from '@/lib/litegraph/src/litegraph'
 import type {
   IBaseWidget,
   IWidgetOptions
@@ -14,6 +15,7 @@ import type {
 import type { InputSpec } from '@/schemas/nodeDef/nodeDefSchemaV2'
 import { useDomWidgetStore } from '@/stores/domWidgetStore'
 import { usePromotionStore } from '@/stores/promotionStore'
+import { useWidgetValueStore } from '@/stores/widgetValueStore'
 import { generateUUID } from '@/utils/formatUtil'
 
 export interface BaseDOMWidget<
@@ -150,6 +152,18 @@ abstract class BaseDOMWidgetImpl<V extends object | string>
     this.callback?.(this.value)
   }
 
+  override setNodeId(nodeId: NodeId): void {
+    // Capture the DOM-resolved value before registration, since the base class
+    // registers _state.value which is undefined for DOM widgets (their value
+    // lives in the DOM element / options.getValue).
+    const resolvedValue = this.value
+    super.setNodeId(nodeId)
+    const graphId = this.node.graph?.rootGraph.id
+    if (!graphId) return
+    const state = useWidgetValueStore().getWidget(graphId, nodeId, this.name)
+    if (state) state.value = resolvedValue
+  }
+
   get margin(): number {
     return this.options.margin ?? BaseDOMWidgetImpl.DEFAULT_MARGIN
   }