Both settings are correct. The one on the support page, imap.mail.me.com is used if someone is setting up their email client manually. If you allow the Mail app to set up iCloud automatically it will pick a specific mail server using criteria set by Apple. Mine picked p08-imap.mail.me.com. I thing it may decide which server to use based on your geographic location.
Back to your problem. In preferences under the General tab there is a setting for the time interval that Mail will use for checking for new mail on the server. It says, Check for new messages: and there is a pull-down menu. Make sure you have it set to an appropriate interval. I'm a little impatient and have a very fast Internet connect so I set mine to, Every 5 minutes.