My Android ver is 4.4.2
Trying to manage the contacts list, the options are as follows:
People => Me => Contacts to Display => Customize => Define Custom View
and then it lists, as the only options, my accounts: Skype, e-mail, Google+ (which I didn't activate, but they list it anyway).
My only choices are to include or exclude each account as a whole, except on the e-mail accounts, where there is option to select My Contacts (Current selected), Starred, Friends, Family, CoWorkers, All Others. No option to select Phones only.
(This is a Motorola Phone, sold while Motorola was a Google company; so I believe this is stock Android - no Motorola customizations)
My Android 4.4.2 People app is the same as yours. I think the designers' thought is that People is where you go for complete contact information; you go to the dialer when you want to make a call. If you follow that path you'll find it doesn't matter how your contacts display in People: only contacts with phone numbers display when you search for a contact within the dialer.