You cleared what the Contacts know but that KEYPAD is the "Phone" app. I do not have a Samsung to tell you exactly what the app name is but here it's always been "Phone."
If you wish, head to settings, then applications (on my phone that's "Settings" and then "Apps") and find the Phone app. Here I had to use the All tab in Apps to find the Phone app. There are clear Cache and Data but this varies so clear the data and cache for the Phone app, reboot and test.
So I factory reset my Galaxy S3... again.
What I want to know is how to get rid of contacts that are persistently staying on my S3 without showing up in the contacts app itself.
What I mean is that I deleted my contacts, so for all options of displaying contacts there are no contacts. However, I go to the keypad and type a number... let's use 2 as an example. 9 contacts show up. All contacts I no longer want.
Without rooting the phone, what can I do about this?
(Samsung phone call says do a hard factory reset using the 3 buttons)

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