## 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)
