We study morphological transitions of droplets on a structured substrate containing two circular lyophilic domains for arbitrary domain and substrate wettabilities. We derive the stability criterion that at least one of the droplets must be pinned at the domain boundary with a contact angle smaller than π/2. This determines seven classes of stable or metastable droplet morphologies of the system. We present a complete classification of stability and metastability of these morphologies as a function of three control parameters as provided by the total droplet volume, substrate wettability, and domain wettability. We find different types of morphological transitions at the stability boundaries: (i) depinning transitions of the contact lines, (ii) symmetry-breaking transitions, where the two droplets acquire different volumes, and (iii) dewetting transitions, where one domain dewets and one of the droplets disappears. We find that depinning transitions of two droplets become discontinuous between two universal values of substrate wettability. Furthermore, below a critical domain wettability, one domain always dewets irrespective of the total volume. We discuss experimental realizations and applications of our results for controlled switching between observed wetting morphologies.