There is no comprehensive list in the format you ask about; you will probably want to look at the original papers. Searching for "exotic" and "4-manifold" on Mathscinet gives > 100 responses, and probably there are other phrases you could use for such a search. There are a number of examples listed in the book of Gompf and Stipsicz, but the subject has progressed quite a bit since that book was published.

The use of handle pictures in describing exotic 4-manifolds is not quite what you are suggesting, though. Typically, an exotic manifold would be described by some other construction (log transforms, fiber sums, knot surgery, rational blowdown, etc.) which enables you to compute a Seiberg-Witten or other gauge-theory invariant. The hard part is typically doing the right sequence of such operations in order to get the right topological type; even computing the fundamental group may be hard. It is usually possible, although often requires considerable effort, to draw a handle picture afterwards. This might be used to uncover some interesting property of the exotic manifold. But I don't know of any examples (at least any simply-connected ones) where one starts with an interesting handle diagram and then deduces exoticity. The point is that one still doesn't know how to effectively compute gauge-theory invariants from the handle picture.