Does the MTA (Postfix, Exim, etc.) get installed seperate from the mail server

Solution 1:

The problem you're having is that "mail server" is an imprecise term. Some people take it to mean "MTA" (because that's all you need to have a server that handles mail), whilst others take it to mean a server that receives mail and stores it for users to manipulate, while still other people have other definitions entirely.

The four software packages you list are, indeed, MTAs, and have little-to-no other functionality provided. Zimbra, et al are what I would call something like "mail service suites", but really there isn't any fully-accepted terminology. It's best to do your own investigation into what a particular program, package or suite does rather than try to rely on imprecise terminology to gauge suitability. (See also: "cloud").