app-server: Return codex home in initialize response (#15689)

This allows clients to get enough information to interact with the codex
skills/configuration/etc.

codex-rs/app-server-protocol/schema/json/v1/InitializeResponse.json

@@ -1,6 +1,20 @@
 {
   "$schema": "http://json-schema.org/draft-07/schema#",
+  "definitions": {
+    "AbsolutePathBuf": {
+      "description": "A path that is guaranteed to be absolute and normalized (though it is not guaranteed to be canonicalized or exist on the filesystem).\n\nIMPORTANT: When deserializing an `AbsolutePathBuf`, a base path must be set using [AbsolutePathBufGuard::new]. If no base path is set, the deserialization will fail unless the path being deserialized is already absolute.",
+      "type": "string"
+    }
+  },
   "properties": {
+    "codexHome": {
+      "allOf": [
+        {
+          "$ref": "#/definitions/AbsolutePathBuf"
+        }
+      ],
+      "description": "Absolute path to the server's $CODEX_HOME directory."
+    },
     "platformFamily": {
       "description": "Platform family for the running app-server target, for example `\"unix\"` or `\"windows\"`.",
       "type": "string"
@@ -14,6 +28,7 @@
     }
   },
   "required": [
+    "codexHome",
     "platformFamily",
     "platformOs",
     "userAgent"