you have the wrong order there
and 2 servers mean 2 different ip's and 2 different names so they can be alive at same time.
keep old server with old name
1 - keep old server with old name
2 - create new server with new name.
3 - install SQL Server now on new server with new name ( sql server default instance will use the hostname and that is what is needed ).
4 - apply login / server security scripts.
5 - script and apply msdb jobs, linked servers from old server to new server.
- - if you have scripts/jobs that update other servers make sure that you change them to point to test servers before running jobs
- - change connection strings as required for testing
6 - restore full backups from old server to new one (copy only backup so log chain on old server is not lost)
7 - use script to restore all backup files to new server.
9 - test
9 - do steps 4 to 7 again
10 - test again
do 9 and 10 as many times as required
11 - go live date
12 - restore full backups from old server
13 - apply connection string changes to all systems affected
14 - final test
15 - signoff
16 - shutdown old server