If you want to do it on the same database instance then take a complete cold Backup yr database(Please do IT). Follow the above procedures Except there is no need to Copy yr datafiles.
I have made little modifications to the above steps
1. Alter database backup control file to trace/xxx.ora. (say xxx.ora)
2. shutdown immediate or normal
3. edit xxx.ora and change the name of yr database and then save as xxx.sql - you need to use startup nomount to run xxx.sql. (delete startup ... from xxx.ora)
edit your initSID file - new db_name and proper destination of your controlfiles...
4. Change ORACLE_SID=Parameter
5.start svrmgrl (or svrmgr23)
6. connect internal/password
7. startup nomount pfile = your initSID file
8. you should be able to open your db with new name.
There Nothing You cannot Do, The problem is HOW.