I think you confused something here. The Phone app shows only contacts, which have a phone number. But: They are still contacts (the phone numbers need a name, don’t they? ;)).
Inside the Contacts app: After choosing “Import/Export” you first have to select from where to export contacts, then where to save them and then, which contacts to export. You could of course tick only the contacts you want to export to your SIM card.
What I would probably do, is to log in to GMail and see if you can specify, which contacts should be synced to your phone.