To tell SQL it has been renamed, you need to run the following.
sp_helpserver select @@servername go sp_dropserver 'OLDSNAME' go sp_addserver 'NEWNAME','local' go sp_helpserver select @@servername go
Remember you will need to restart the SQL Server service after executing this. You could even do it automatically using this Rename SQL Installation script from back in the day. It will detect the names automatically.