[FS] Ensure parent node is always updated in rename(), regardless of backend (#21964)

Doing the update in the core logic avoids each backend needing to do it,
and avoids the risk of a backend forgetting (as was the case with NODEFS,
which this fixes).

Author: SiobhyB

src/library_fs.js

@@ -812,6 +812,9 @@ FS.staticInit();` +
       // do the underlying fs rename
       try {
         old_dir.node_ops.rename(old_node, new_dir, new_name);
+        // update old node (we do this here to avoid each backend 
+        // needing to)
+        old_node.parent = new_dir;
       } catch (e) {
         throw e;
       } finally {

src/library_memfs.js

@@ -198,7 +198,6 @@ addToLibrary({
         old_node.name = new_name;
         new_dir.contents[new_name] = old_node;
         new_dir.timestamp = old_node.parent.timestamp;
-        old_node.parent = new_dir;
       },
       unlink(parent, name) {
         delete parent.contents[name];

src/library_proxyfs.js

@@ -108,7 +108,6 @@ addToLibrary({
         try {
           oldNode.mount.opts.fs.rename(oldPath, newPath);
           oldNode.name = newName;
-          oldNode.parent = newDir;
         } catch(e) {
           if (!e.code) throw e;
           throw new FS.ErrnoError(ERRNO_CODES[e.code]);