Well, you have to use the shortcode. The job of the audio post format isn’t to turn a URL into a player, it’s job is merely to change the appearance of a post so as to mark that the post is or contains audio.
“Audio embed code or URL”: where exactly did you see this?
What I can say is
a) An mp3 URL will automatically be turned into a player if the mp3 has been uploaded to your blog; if it’s hosted elsewhere you need the shortcode.
b) What you saw in the post is wrong anyway: pasting the URL should produce nothing, or the actual URL, or a link, or a player – anything other than code!