Save contacts to phone not Google account On my new ASUS ZenFone I could not figure out how to save contacts just to my phone.
The default app should be Google Contacts and despite some suggestions even here it does not seem possible. There is no option to choose where to save new contact. There is just Default account for new contacts and it is set to my Google account, as there is no other account to by set.  

EDIT: I decided to use MyLocalAccount app to fix this problem with Contacts app. But now I have the same problem with Phone app, which is showing all contacts in its Contacts tab.
I probably could live without ever going to this tab, but it would be nice if there is a way to do it.
You should try to use a 3rd party Contacts app. I have seen similar behavior in Motorola contacts app. They left no option to store contacts locally to the device. 
Most likely using the Play Store version of Google Contacts will help. If not, try something else. There are ample contacts app and I do not wanna be partial about a specific app.