Setting up CNAME record with an API served on TLD fails.

I'm confused. I host my API at my TLD. Using the apigee provided url works fine and it does neatly track all the request.

But I created a CNAME record (api.phonescomplete.com -> phonescomplete.apigee.com -> phonescomplete.com) which doesn't appear to work.

To my understanding a CNAME record is a bit like a sym link and I it points to the correct IP address. As a matter of fact when doing an api call on api.phonescomplete.com apigee inserts a couple of headers:
X-APIGEE-ERROR: internal-error
X-APIGEE-STATUS: failure

and body:
Requested Domain api.phonescomplete.com not registered

Which is correct of course. I didn't register api.phonescompleten.com, I registered phonescomplete.com.
I figured maybe I needed to change the CNAME field in the apigee settings of my API but I can't update that field. The error:
Could not update the API.

Not very helpful I'm afraid.

Not sure what I'm doing wrong here...