Support selection of app inputs and outputs from vue mode (#9259)

- The input and output indicators are now plugged directly into the
`LGraphNode.vue` template. Care was taken to make implementation to have
low cost for performance and complexity when not in app mode setup.
- Context menu event handlers are added to each widget in vue mode
instead of resolving the target widget of an event
- Swap the nodeId passed by `useGraphNodeManager` to not include the
locator id. This id was never used and was incorrect since it didn't
resolve across nested subgraphs.
- Continued bug fixes for app mode as a whole.

Known issue: There is disparity of nodeId between litegraph (which
references the widget in the root graph) and vue (which promotes the
original widget). Efforts to reconcile are ongoing.

┆Issue is synchronized with this [Notion
page](https://www.notion.so/PR-9259-Support-selection-app-inputs-and-outputs-from-vue-mode-3136d73d365081ae8e56e35bf6322409)
by [Unito](https://www.unito.io)

---------

Co-authored-by: pythongosssss <125205205+pythongosssss@users.noreply.github.com>

src/components/LiteGraphCanvasSplitterOverlay.vue

@@ -155,7 +155,7 @@ const unifiedWidth = computed(() =>
 
 const { focusMode } = storeToRefs(workspaceStore)
 
-const { isSelectMode } = useAppMode()
+const { isSelectMode, isBuilderMode } = useAppMode()
 const { activeSidebarTabId, activeSidebarTab } = storeToRefs(sidebarTabStore)
 const { bottomPanelVisible } = storeToRefs(useBottomPanelStore())
 const { isOpen: rightSidePanelVisible } = storeToRefs(rightSidePanelStore)
@@ -163,7 +163,9 @@ const showOffsideSplitter = computed(
   () => rightSidePanelVisible.value || isSelectMode.value
 )
 
-const sidebarPanelVisible = computed(() => activeSidebarTab.value !== null)
+const sidebarPanelVisible = computed(
+  () => activeSidebarTab.value !== null && !isBuilderMode.value
+)
 
 const sidebarStateKey = computed(() => {
   return unifiedWidth.value