I’m confused about the difference between the php IMAP extension and and IMAP server.
The php version on your server has the php extension for accessing IMAP mail servers. However, this extension only allows php to connect to an IMAP server. It does not actually provide an IMAP server. By default most accounts are not configured for IMAP mail service. We can upgrade you to a mail server that does (no extra charge) – contact support to have the IMAP mail server upgrade performed.
If you simply want to connect to other IMAP mail servers, you do not need to do anything. If you want to connect to your own mail via imap/webmail you’ll need to upgrade to an IMAP mail server with us. As a side-note, part of the IMAP conversion process also gives you access to a pre-installed version of roundcube (you are still welcome to do your own roundcube install if you prefer).