Apple Mail seems to always maintain a Sent Messages folder for emails sent from Mac with IMAP mail. Even if you manually change Sent Messages to some other name, Apple Mail creates a new Sent Messages folder on IMAP server.
To put sent messages in default Sent folder, click and highlight Sent Mailbox, go to Mailbox – Use Mailbox for – Sent

Windows Live Mail
Stupid Live Mail by default creates another set of folders outside the root INBOX. Cyrus only allows folders under INBOX. Solution: in account properties, IMAP tab, set Root folder path to INBOX, or Inbox. Both seem to work. And specify corresponding folders on server for Sent Items, Drafts path, Deleted Items, etc. These folder names are case sensitive, while INBOX is not.

“For Cyrus servers, all users’ folders must be contained in the Inbox folder. ”

It seems like Outlook 2013 will have the same problem, yet not mentioned in the official CMU tutorial. CMU’s tutorial seems to suggest that everything will work automatically, and to some extent that is the case, incoming and outgoing servers are automatically detected. But, after initial setup, Outlook 2013 showed a folder structure similar to the (incorrect) one I saw in Live Mail, in which Inbox seems to be a standalone mailbox alongside, and Draft, Sent, Trash are subfolders of Inbox.
A quick peek in the account settings you can find a field for Root folder path under More Settings – Advanced tab.