Two-phase commit (2PC) is widely used in distributed databases to ensure the atomicity of distributed transactions. However, 2PC has two limitations. First, it requires two eager log writes on the critical path, which incurs significant latency. Second, when a coordinator fails, a participant may be blocked waiting for the coordinator's decision, leading to indefinitely long latency and low throughput. 2PC was originally designed for a shared-nothing architecture. We observe that the two problems above can be addressed in an emerging storage disaggregation architecture which provides compare-and-swap capability in the storage layer. We propose Cornus, an optimized 2PC protocol for Cloud DBMS with Storage Disaggregation. We present Cornus in detail with proofs and show how it addresses the two limitations in 2PC. We also deploy it on real storage services including Azure Blob Storage and Redis. Empirical evaluations show that Cornus can achieve up to 1.9x speedup in latency over 2PC.