Put bluntly, the great listening public requires to have the tune laid recognisably on its ears before it can appreciate having that tune reshaped. Carter often dispensed with such formality and moved directly to her inventions on the melody without an unembellished theme statement. Undoubtedly one of the most important jazz singers of the age, Carter was also a great teacher of musicians with a dominant belief in the spirit of jazz rather than its definition. It's not technical. It's a feeling you receive from a performer.
Most vocalists sing the melody with support from a backing group, perhaps a trio or a big band.
Carter, but not Carter who, like Billie Holiday, sang from within whatever group she was working with, rather than just using it as a platform. I think it used to bother musicians that I didn't sing the melody. But I think you should do a tune the way you feel. Because it's been done straight for the last 50 years. And there's ten thousand singers out there who will sing it straight, who can't improvise, who don't even know how.