docs: add widget serialization reference (widget.serialize vs widget.options.serialize) (#9102)

Documents the distinction between `widget.serialize` (workflow
persistence, checked by `LGraphNode.serialize()`) and
`widget.options.serialize` (API prompt, checked by `executionUtil.ts`).

These share a property name but live at different levels of the widget
object and are consumed by different code paths — a common source of
confusion when debugging serialization bugs.

Includes:
- Explanation of both properties with code references
- Permutation table of the 4 possible combinations with real examples
- Gotchas section covering the `addWidget` options bag behavior and
`PrimitiveNode` dynamic widgets
- Reference added to `src/lib/litegraph/AGENTS.md`

Context: discovered while debugging #1757 (PrimitiveNode
`control_after_generate` lost on copy-paste).

┆Issue is synchronized with this [Notion
page](https://www.notion.so/PR-9102-docs-add-widget-serialization-reference-widget-serialize-vs-widget-options-serialize-30f6d73d365081cd86add44bdaa20d30)
by [Unito](https://www.unito.io)

---------

Co-authored-by: GitHub Action <action@github.com>

src/lib/litegraph/AGENTS.md

@@ -6,6 +6,10 @@
 - Avoid repetition where possible, but not at expense of legibility
 - Prefer running single tests, not the whole suite, for performance
 
+## Widget Serialization
+
+See `docs/WIDGET_SERIALIZATION.md` for the distinction between `widget.serialize` (workflow persistence) and `widget.options.serialize` (API prompt). These are different properties checked by different code paths — a common source of confusion.
+
 ## Code Style
 
 - Prefer single line `if` syntax for concise expressions