Communication Log - REST API, different Message Type

How do I add something to the Communication log using a Message Type other than email? The SQL table CommunicationMessageTypeRef has 4 different options, and I want to use something other than "Email" but I can't figure out which field in which call to insert the different value. Everything I try just shows Email on the output. I have no problems (now) writing to the log (3 POST calls but it's there), just can't get that value correct.

Thanks.