LMTP is a small modification of SMTP with a per-recipient response code - but I'm not sure what it's intended for. It isn't used on the internet. It's sometimes used as IPC within a mail server.
LMTP is for delivering to an IMAP (or JMAP) mailbox, generally. We also have a plugin for that. But it has certain flaws for exposing it to the internet (I just forget what they are right now).
There's no such thing as an IMAP mailbox. Mail receiving is split down the middle. Your SMTP server stores the email in a folder and your IMAP server provides access to that folder.
For context, I'm the author of Haraka, a popular open source SMTP server.
SMTP servers don't necessarily store anything. They have two responsibilities: Receiving and delivering email. What they do with the emails they receive is up to the configuration.
