We classify rank 2 cluster varieties (those whose corresponding skew-form has rank 2) according to the deformation type of a generic fiber U of their X-spaces, as defined by Fock and Goncharov. Our approach is based on the work of Gross, Hacking, and Keel for cluster varieties and log Calabi-Yau surfaces. We find, for example, that U is "positive" (i.e., nearly affine) and either finite-type or non-acyclic (in the cluster sense) if and only if the monodromy of the tropicalization of U is one of Kodaira's matrices for the monodromy of an ellpitic fibration. In the positive cases, we also describe the action of the cluster modular group on the tropicalization of U.