Order of phone number types when adding a new contact
It appears to be related to the service you are using.

Open Settings, find Contacts, and see what it's set to for Default Account. Apparently the service provides the list of account types.

Mine is set to Exchange, so when I create a new contact, and add a phone, it defaults to Home Fax.

If I change the Default Account to iCloud, it defaults the phone type to Phone.