The phone has contacts 'on the phone' which is a separate data store from an external server which you sync with, and the same problem occurs whether you sync with Traveler, Exchange, GMail or anything else.
From the Contacts app, when you touch 'Groups' at the top of the view you will see the list of data stores. For example, I sync with Traveler and with GMail so I see
All on My iPhone (note - these do not sync with anything)
You can set in your preferences which data store is the default for new contacts, or you can navigate to that data store and create a new contact there. But once created there is no moving it.
This is a major flaw in the iPhone app as far as I'm concerned (by contrast you can move a calendar entry from one calendar to another).
There are only two ways I can think of
- try to sync the contacts on the phone with a desktop app or cloud contacts app (google, apple) via iTunes - which should sync the contacts 'on my Phone' but not the Traveler contacts. Then you can export them from there and import them to Notes.
Alternatively send each one from the iPhone to your Notes email using the 'Share Contact' button then open the vcard in Notes and add it to your contacts.
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