IPhone's Phone contacts vs. iPhone's Contact contacts!

Question

I'm really confused by the difference between the contacts I see in the iPhone's phone side of things and the "official" Contacts. They are called the same thing and yet they are different, which is really dumb IMHO. For instance, if I create a new contact (on the phone side of the device) based on the phone number of someone who's called me, it doesn't appear in my official Contacts list. What's the point of maintaining two separate lists of contacts within one iPhone? The consequence is that if I look up that individual in my (official) contacts, he's not there, so I'm constantly having to remember how I created a contact in the first place to know where to look. Am I missing something?

Answer 1

Thanks again. In response to your questions:
I believe I am saving to iCloud, but I'm not clear about your Settings path: in the Contacts section in Settings there is no "Default Account", but there is in the Mail section, which is presumably irrelevant (this is anyway set to iCloud).
Not sure what you mean by "is it selected as a group to show contacts" - kindly explain. When I tap Groups in my Contacts, "All Contacts" is ticked, and the (non-editable) heading at the top says ICLOUD.
Yes, I tap Done.
Just to reiterate: the new contacts I create based on incoming or outgoing phone calls are not appearing in my Contacts, yet they are registered in the Recents page and they appear under Info when I tap the Info icon. When I create the new contact, I see the little activity wheel turning to indicate that the information is being sent somewhere.