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/codex_app_server_protocol.schemas.json

@@ -1,6 +1,10 @@
 {
   "$schema": "http://json-schema.org/draft-07/schema#",
   "definitions": {
+    "AbsolutePathBuf": {
+      "description": "A path that is guaranteed to be absolute and normalized (though it is not guaranteed to be canonicalized or exist on the filesystem).\n\nIMPORTANT: When deserializing an `AbsolutePathBuf`, a base path must be set using [AbsolutePathBufGuard::new]. If no base path is set, the deserialization will fail unless the path being deserialized is already absolute.",
+      "type": "string"
+    },
     "AdditionalFileSystemPermissions": {
       "properties": {
         "read": {
@@ -2305,6 +2309,14 @@
     "InitializeResponse": {
       "$schema": "http://json-schema.org/draft-07/schema#",
       "properties": {
+        "codexHome": {
+          "allOf": [
+            {
+              "$ref": "#/definitions/v2/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"
@@ -2318,6 +2330,7 @@
         }
       },
       "required": [
+        "codexHome",
         "platformFamily",
         "platformOs",
         "userAgent"