Here's the problem: there isn't "one type" of calvinist.
Within calvinism, you have sub-sets: those that ascribe to lapsanarianism, which also has subsets ranging from supra to ante and infra to post; 4-point calvinists; neo-orthodox; neo-calvinists; hyper-calvinists...and I'm sure there's one or two others; and each of those has sub-sets within them regarding certain points.
You ask why a calvinist would be a missionary? A hyper-calvinist would not, because he would be exactly as you describe: someone who believes that God will save all who will be saved. A typical calvinist, however, says that we have no idea who God has pre-ordained, and that faith comes by hearing, and hearing by the Word; so therefore, go out an preach, and rejoice when God saves those He has chosen.
It's like using "fundamentalist" to describe terrorists. Most terrorists may be fundamentalists; but by no means are all or even most fundamentalists terrorists. These kind of logic errors lead to misunderstanding, which causes dissension and strife.
The reason the labels are causing such a fuss is because there is a tendency to lump all calvinists into one group, and all "non-cals" into another group, without regard for specific beliefs; thus causing the error of generalization. Tying that back to my opening paragraph: the reason we make the generalization error is because we don't know that there are different beliefs within an overall grouping, so we make assumptions about certain people based on the label. Like hearing that someone is a fundamentalist, and automatically thinking, "oh, another Timothy McVeigh."
Someone asked in another thread about non-cal authors, with no regard for the multitude of different types of beliefs that range from arminian to lutheran to pelagian, etc. Here on this thread, we see people talking about calvinists with no regard for whether the calvinist in question is lapsanarian, or neo-calvinist, etc.
So, my suggestion? Lose the labels, talk the theology.