Make CaretBase a DisplayItemClient.

third_party/WebKit/LayoutTests/paint/invalidation/textarea-caret-expected.txt

 {
   "name": "Content Root Layer",
   "bounds": [800, 600],
   "children": [
     {
       "name": "LayoutView #document",
       "bounds": [800, 600],
       "contentsOpaque": true,
       "drawsContent": true,
       "paintInvalidations": [
         {
-          "object": "LayoutTextControl INPUT id='target'",
-          "rect": [5, 5, 54, 25],
+          "object": "LayoutTextControl TEXTAREA id='editor'",
+          "rect": [7, 7, 183, 40],
+          "reason": "subtree"
+        },
+        {
+          "object": "LayoutText #text",
+          "rect": [9, 11, 179, 16],
           "reason": "subtree"
         },
         {
           "object": "LayoutBlockFlow DIV id='inner-editor'",
-          "rect": [11, 11, 42, 13],
+          "rect": [11, 11, 175, 16],
           "reason": "subtree"
         },
         {
+          "object": "LayoutTextControl TEXTAREA id='editor'",
+          "rect": [9, 30, 164, 15],
+          "reason": "scroll"
+        },
+        {
+          "object": "LayoutTextControl TEXTAREA id='editor'",
+          "rect": [173, 30, 15, 15],
+          "reason": "scroll"
+        },
+        {
           "object": "LayoutText #text",
-          "rect": [11, 11, 42, 13],
-          "reason": "subtree"
+          "rect": [186, 10, 2, 18],
+          "reason": "invalidate paint rectangle"
         }
       ]
     }
   ],
   "objectPaintInvalidations": [
     {
-      "object": "LayoutBlockFlow HTML",
-      "reason": "selection"
-    },
-    {
-      "object": "LayoutBlockFlow BODY",
-      "reason": "selection"
-    },
-    {
-      "object": "LayoutTextControl INPUT id='target'",
+      "object": "LayoutTextControl TEXTAREA id='editor'",
       "reason": "subtree"
     },
     {
+      "object": "LayoutTextControl TEXTAREA id='editor'",
+      "reason": "scroll"
+    },
+    {
+      "object": "HorizontalScrollbar",
+      "reason": "scroll"
+    },
+    {
       "object": "LayoutBlockFlow DIV id='inner-editor'",
       "reason": "subtree"
     },
     {
       "object": "RootInlineBox",
       "reason": "subtree"
     },
     {
-      "object": "HorizontalScrollbar",
-      "reason": "scroll"
-    },
-    {
       "object": "LayoutText #text",
       "reason": "subtree"
     },
     {
-      "object": "InlineTextBox 'test test test'",
+      "object": "InlineTextBox '------------------------------------------------------------'",
       "reason": "subtree"
+    },
+    {
+      "object": "Caret",
+      "reason": "invalidate paint rectangle"
     }
   ]
 }