That article describes push as how it works for apps. For the iOS Mail program it works slightly differently. For mail servers that support push mail in the iOS Mail app (iCloud, Yahoo, Outlook, etc), when a new mail arrives at the server, the server sends a message to Apple's Push Notification server as described in the article, but it doesn't get sent to the iOS device as a normal push notification.
When a mail server push notification comes in, it triggers the iOS Mail app to go out and fetch all new Mail for that server's inbox via IMAP. That's why when push mail is working correctly, you can actually put the phone in airplane mode after mail is pushed to the iOS device and open the app and read the mail. This is actually easy to see if the Mail app is already open when new mail arrives as you'll see it download all the new Mail for that server.
That differs to how the article describes it, which is sending a push notification containing a brief summary of the email. That's how third party apps like Yahoo Mail and Gmail work, but the native Mail app is special.