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.
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.