What is the appropriate name for this dataset? I'll assume most of us know about cross-validation and are accustomed to using Training and Test sets. However, I've also seen references to a third set that you use to test a completed model. It's the test to see how your model does against a dataset that it hasn't seen before; helps determine if your model has been overly-fit to the training data. 

I have seen this dataset referred to everything from "the third dataset", "the wild dataset", "the real-world dataset". So what is the appropriate name for this third dataset?
Well, I would not call this "Test" set you defined as "Test"! I would say: Train, Validation, Test. And this "Test" is the dataset that your model has not seen yet. This is pretty standard even explained in details in Andrew Ng courses like this very useful and important lesson on splitting data and the significance of their distributions.