Q: Missing Contacts after backup

If you old phone was still connected to your accounts and you were manually deleting them, then you successfully deleted them form your old phone, your synced account, and subsequently your new Phone.

To remove everything form your old phone, you should go to Settings > General > Reset > Erase All Content and Settings.  This will remove the accounts and other stuff from your phone without deleting the information from your synced accounts.

Since you have more than likely removed all of the contacts form a synced account, you will need to contact whoever owns the accounts you sync with.  Most people use iCloud primarily.  You can check in Settings > iCloud on your new iPhone.  If contacts is turned on, then you will want to contact Apple Support to see if they can help you recover your contacts you accidently deleted.

You can also check Settings > Mail, Contacts, Calendars and view all of your accounts (including iCloud) to see which ones show Contacts as being turned on, as any of these accounts could have been hosting your deleted contacts (but it was more than likely iCloud or your primary email account if those have contacts turned on).

How to contact Apple about your iCloud account: http://support.apple.com/kb/he57

No problem.  Glad I could help at least shine some light on what happened