It was proved in [L. Noakes and L. Stoyanov, Proc. Amer. Math. Soc., 143 (2015), pp. 3879-3893] that obstacles K in R^d that are finite disjoint unions of strictly convex domains with C^3 boundaries are uniquely determined by the traveling times of billiard trajectories in their exteriors and also by their so-called scattering length spectra. However, the case d = 2 is not covered by that article. In the present paper we give a separate and different proof of this result for the case when d = 2.